Two-dimensional Ice Research Articles

A consistent phase-field model for three-phase flows with cylindrical/spherical interfaces

A phase-field model for three-phase flows with cylindrical/spherical interfaces is established by combining the Navier-Stokes (NS), the continuity, and the energy equations, with an explicit form of curvature-dependent modified Allen-Cahn (AC) and Cahn-Hilliard (CH) equations. These modified AC and CH equations are proposed to solve the inconsistency of the phase-field method between flat and curved interfaces, which can result in “phase-vanishing” problems and the break of mass conservation during the phase-changing process. It is proved that the proposed model satisfies the energy dissipation law (energy stability). Then the icing process with three phases, i.e., air, water, and ice, is simulated on the surface of a cylinder and a sphere, respectively. It is demonstrated that the modification of the AC and CH equations remedies the inconsistency between flat and curved interfaces and the corresponding “phase-vanishing” problem. The evolution of the curved water-air and the water-ice interfaces are captured simultaneously, and the volume expansion during the solidification owing to the density difference between water and ice agrees with the theoretical results. A two-dimensional icing case with bubbles rising is simulated. The movement and deformation of bubbles, as well as the evolution of the interfaces, effectively illustrate the complex interactions between different phases in the icing process with phase changes.

An improved artificial spin ice structure for restoring ice degeneracy

It is known that rare-earth-based pyrochlore oxide may accommodate the well-defined two-in-two-out spin ice state with its tetrahedral unit. Low-energy excitation is argued to favor the highly concerned monopole state which attracts essential attention. However, such an excitation cannot be directly tracked and imaged using advanced characterizations, raising challenges to our understanding of the physics of monopoles. In this work, we propose an improved two-dimensional artificial spin ice structure on the Shastry–Sutherland lattice to restore the degeneracy of realistic pyrochlore systems. Such a structure avoids the deficiency of inequivalent nearest and next-nearest exchanges in the planar quadrate unit, which, however, is equivalent to the tetrahedral unit of realistic pyrochlore oxides. Therefore, this spin ice model restores state degeneracy that is lost in conventional planar artificial spin ice structures, representing an improved simulator of real spin ice systems. Our careful investigations of such improved structures reveal the rich physics of spin ice excitations, including the phase diagram, which allows different ordered phases and interesting critical phase transitions between spin ice phase I and phase II. Energy spectrum analysis suggests that restoration of state degeneracy substantially reduces monopole excitation energy, resulting in a striking monopole emergency at the critical point. Furthermore, the emergent spin dimer phase in this improved model allows high-density monopole excitations and exhibits high-correlated monopole fluid states.

Two-dimensional bilayer ice in coexistence with three-dimensional ice without confinement

Icing plays an important role in various physical-chemical process. Although the formation of two-dimensional ice requires nanoscale confinement, two-dimensional bilayer ice in coexistence with three-dimensional ice without confinement remains poorly understood. Here, a critical value of a surface energy parameter is identified to characterize the liquid-solid interface interaction, above which two-dimensional and three-dimensional coexisting ice can surprisingly form on the surface. The two-dimensional ice growth mechanisms could be revealed by capturing the growth and merged of the metastable edge structures. The phase diagram about temperature and pressure vs energy parameters is predicted to distinguish liquid water, two-dimensional ice and three-dimensional ice. Furthermore, the deicing characteristics of coexisting ice demonstrate that the ice adhesion strength is linearly related to the ratio of ice-surface interaction energy to ice temperature. In addition, for gas-solid phase transition, the phase diagram about temperature and energy parameters is predicted to distinguish gas, liquid water, two-dimensional ice and three-dimensional ice. This work gives a perspective for studying the singular structure and dynamics of ice in nanoscale and provides a guide for future experimental realization of the coexisting ice.

Modeling two-dimensional ice shape based on fractal interpolation

The ice accretion data obtained from ice wind tunnel tests reveal a multiscale structure and rough surface. In the follow-up aerodynamic evaluation of icing airfoils, simplified two-dimensional ice shapes are generally used as substitutes, but this simplification changes the aerodynamic effects of the original ice shape. Therefore, finding a simple and effective two-dimensional ice shape simulation method is urgent. Due to the self-similarity characteristics of ice shape surfaces, fractal interpolation is proposed for ice shape simulation. First, the geometric characteristics of the ice shapes are analyzed to determine interpolation points, and an iterative function system is constructed for interpolation simulation. Considering the influence of various characteristics of ice shapes on aerodynamics, interpolation parameters are limited to simulating more realistic ice shapes. High-order numerical simulation methods were utilized to numerically simulate and analyze the aerodynamic characteristics of icing airfoil while also verifying the feasibility of fractal interpolation for simulating ice shapes. The analysis revealed that this method could effectively simulate ice profiles of various feature scales with minimal ice shape data. These simulated shapes closely resemble real ice formations and maintain the original aerodynamic characteristics of the icing airfoil. This method can be used to improve the computational accuracy of ice accretion codes and provide improvement strategies for complex ice shape prediction; thus, this method has great application prospects in engineering.

Two rhombic ice phases from aqueous salt solutions under graphene confinement.

Water exhibits rich ice phases depending upon its respective formation conditions, and in particular, the two-dimensional ice with nonhexagonal symmetry adsorbed on solids relates to the exceptional arrangement of water molecules. Despite extensive reporting of two-dimensional ice on various solid surfaces, the geometry and thermodynamics of ice formation from an aqueous salt solution are still unknown. In this Letter, we show the formation of single- and two-phase mixed two-dimensional rhombic ice from aqueous salt solutions with different concentrations under strong compressed confinement of graphene at ambient temperature by using classical molecular dynamics simulations and first-principles calculations. The two rhombic ice phases exhibit identical geometry and thermodynamic properties, but different projections of the oxygen atoms against solid surface symmetry, where they relate to the stable and metastable arrangements of water molecules confined between two graphene layers. A single-phase rhombic ice would grow from the confined saturated aqueous solutions since the previously stable rhombic molecular arrangement becomes an unstable high-energy state by introducing salt ions nearby. Our result reveals different rhombic ice phases growing from pure water and aqueous solutions, highlighting the deciding role of salt ions in the ice formation process due to their common presence in liquids.

The performance of OPC and OPC3 water models in predictions of 2D structures under nanoconfinement.

Nanoconfined water plays an important role in broad fields of science and engineering. Classical molecular dynamics (MD) simulations have been widely used to investigate water phases under nanoconfinement. The key ingredient of MD is the force field. In this study, we systematically investigated the performance of a recently introduced family of globally optimal water models, OPC and OPC3, and TIP4P/2005 in describing nanoconfined two-dimensional (2D) water ice. Our studies show that the melting points of the monolayer square ice (MSI) of all three water models are higher than the melting points of the corresponding bulk ice Ih. Under the same conditions, the melting points of MSI of OPC and TIP4P/2005 are the same and are ∼90K lower than that of the OPC3 water model. In addition, we show that OPC and TIP4P/2005 water models are able to form a bilayer AA-stacked structure and a trilayer AAA-stacked structure, which are not the cases for the OPC3 model. Considering the available experimental data and first-principles simulations, we consider the OPC water model as a potential water model for 2D water ice MD studies.

Formation of a two-dimensional helical square tube ice in hydrophobic nanoslit using the TIP5P water model.

In extreme and nanoconfinement conditions, the tetrahedral arrangement of water molecules is challenged, resulting in a rich and new phase behavior unseen in bulk phases. The unique phase behavior of water confined in hydrophobic nanoslits has been previously observed, such as the formation of a variety of two-dimensional (2D) ices below the freezing temperature. The primary identified 2D ice phase, termed square tube ice (STI), represents a unique arrangement of water molecules in 2D ice, which can be viewed as an array of 1D ice nanotubes stacked in the direction parallel to the confinement plane. In this study, we report the molecular dynamics (MD) simulations evidence of a novel 2D ice phase, namely, helical square tube ice (H-STI). H-STI is characterized by the stacking of helical ice nanotubes in the direction parallel to the confinement plane. Its structural specificity is evident in the presence of helical square ice nanotubes, a configuration unseen in both STI and single-walled ice nanotubes. A detailed analysis of the hydrogen bonding strength showed that H-STI is a 2D ice phase diverging from the Bernal-Fowler-Pauling ice rules by forming only two strong hydrogen bonds between adjacent molecules along its helical ice chain. This arrangement of strong hydrogen bonds along ice nanotube and weak bonds between the ice nanotube shows a similarity to quasi-one-dimensional van der Waals materials. Ab initio molecular dynamics simulations (over a 30ps) were employed to further verify H-STI's stability at 1GPa and temperature up to 200K.

Magnetic monopole free motion in two-dimensional artificial spin ice

Magnetic monopole motion (without its partner in a pair) as a function of external magnetic fields is presented as a fingerprint of freedom of these emergent quasi-particles in a two-dimensional artificial spin ice material. Such freedom, required, for example, for further application in magnetricity, is only possible due to ground-state degeneracy, which causes an anisotropic string energy reduction in rectangular ice geometries, designed to allow highest equidistance among lateral and diagonal neighbor nanomagnets. By measurements of magnetic force microscopy, we observe magnetic monopole creation, transport, and annihilation in such an arrangement, where monopoles are free to move throughout large regions of the sample, since they are not permanently bound by the string bonds, as commonly observed in conventional artificial spin ice systems. Instead, monopoles leave a wake of elongated tensionless string, which is connected to superficial charges or charges pinned to the array borders.

Theoretical investigations of the vibrational spectra and hydrogen bond vibrations of two-dimensional ice I

In 2020, experimental observations of a stable two-dimensional (2D) ice I structure on a gold substrate were reported. As an atomic-level material, 2D ice has potential applications in many fields. However, the vibrational spectra of 2D ice I, including its infrared (IR) absorption, Raman scattering, and inelastic neutron-scattering spectra, have not been collected, owing to the complexity of the experimental conditions required to do so. We used first-principle density functional theory calculations to model 2D ice I and proved that it can stably exist without a gold substrate. We also simulated the vibrational phonon density of states and the IR and Raman spectra of 2D ice I. The vibrational frequencies in the translational band of 2D ice I were lower than those of 3D ice, owing to the hydrogen bonds of the former being weaker than those of the latter. The dynamic process analysis of hydrogen bonds showed that the vibrational modes of 2D ice I are quite different with that of 3D ice showing a different vibrational spectrum.

Numerical modeling of ice control on the Albany River, Ontario, Canada

The Kashechewan First Nation (KFN) community is located on a floodplain of the north branch of the Albany River, on the west coast of James Bay. Each spring, the community faces the uncertain prospect of evacuation due to ice-jam flooding, the most severe of which occurred in 2006. A study was conducted to develop long-term options for reducing the flood risk to the community, which included dams, dykes, pier and weir type ice-control structures, and conveyance channels. This paper presents a numerical model study on the Albany River in Ontario, Canada to develop ice control as a potential means for mitigating ice-jam flooding at the KFN community using the two-dimensional ice dynamics model DynaRICE. A method was developed to simulate pier type ice-control structures within the model such that the physical channel conditions, flow conditions, and combinations of ice control could be evaluated. The model allowed detailed simulation of ice retention, resulting inundation, and the forces on the individual piers. To optimize the design of the piers, including pier spacing, height, and size, various combinations of ice, flow, and pier conditions were simulated. Numerical modeling facilitated a comparative examination of flood risk reduction to the community from the various ice control options.

Observing growth and interfacial dynamics of nanocrystalline ice in thin amorphous ice films

Ice crystals at low temperatures exhibit structural polymorphs including hexagonal ice, cubic ice, or a hetero-crystalline mixture of the two phases. Despite the significant implications of structure-dependent roles of ice, mechanisms behind the growths of each polymorph have been difficult to access quantitatively. Using in-situ cryo-electron microscopy and computational ice-dynamics simulations, we directly observe crystalline ice growth in an amorphous ice film of nanoscale thickness, which exhibits three-dimensional ice nucleation and subsequent two-dimensional ice growth. We reveal that nanoscale ice crystals exhibit polymorph-dependent growth kinetics, while hetero-crystalline ice exhibits anisotropic growth, with accelerated growth occurring at the prismatic planes. Fast-growing facets are associated with low-density interfaces that possess higher surface energy, driving tetrahedral ordering of interfacial H2O molecules and accelerating ice growth. These findings, based on nanoscale observations, improve our understanding on early stages of ice formation and mechanistic roles of the ice interface.

Terahertz electric field induced melting and transport of monolayer water confined in double-walled carbon nanotubes.

Understanding the structures and dynamics of confined water in nanochannels holds great promise for various applications, ranging from membrane separation to blue energy collection. A setting of particular interest is the confined monolayer water within double-walled carbon nanotubes (DWCNTs), which demonstrates rich ice morphologies; however, the dynamics of this peculiar system are still unexplored. In this work, a series of molecular dynamics (MD) simulations reveal that the two-dimensional ice in DWCNTs can be effectively melted by terahertz electric fields but not by static electric fields, exhibiting an interesting ice to vapor-like transition along with extraordinary dynamical behaviors. Specifically, under appropriate field frequency, the water flow presents a sharp increase with the increase in field strength, indicating an excellent gating behavior. These remarkable findings are attributed to the resonance effect between the terahertz electric field and inherent vibration of water hydrogen bonds, causing the water molecules to change from the frozen to super permeation states. The amount of confined water exhibits a sudden reduction, confirming the breakdown of the hydrogen bond network. The distributions of density profiles, hydrogen bond number and dipole orientation demonstrate more details of the water structural change. Furthermore, under a certain field strength, the water flow shows a peculiar maximum behavior with the increase in field frequency, implying frequency optimization for water transport. These findings not only enhance our understanding of the phase transition behavior of water molecules confined within DWCNTs under the influence of a terahertz electric field but also provide a promising avenue for designing innovative nanofluidic devices.

High thermal conductivity and ultralow friction of two-dimensional ice by molecular dynamics simulations

Two-dimensional ice (2D ice) is a relatively new family member of ice polymorphs. While the phase transitions and growth kinetics of 2D ice have aroused intensive investigation, its thermal and mechanical properties are seldom studied. The unique 2D structure of this polymorph suggests that it may possess distinct physical characteristics compared to its bulk counterparts. In this work, the thermal conductivities and friction of 2D ice were systematically studied by molecular dynamics (MD) simulations. It was found that the thermal conductivities of 2D ice show a strong dependence on their length and the extrapolated value of the infinite system can achieve ∼4.7 W/(m∙K), which is notably higher than that of bulk phase of ice Ih ∼1.5 W/(m∙K). Decreasing temperature or applying mechanical strains can both increase the thermal conductivity of 2D ice, which can be explained by structure or phonon analyses. 2D ice also exhibits an ultralow friction of ∼0.24 nN when sliding upon a graphene substrate, lower than that of ice Ih ∼0.70 nN at the same sliding conditions. With the decrease in temperature or the increase in normal load, the friction coefficient of 2D ice exhibits a downward trend, and it could be as low as ∼0.005, indicating superlubricity. The analysis on structural lubricity suggests that ordered but mismatched crystal-crystal contacts have lower friction than disordered case. The results on high thermal conductivity and ultralow friction would guide the potential application of 2D ice in efficient thermal management or design of nanofluidic devices.

Roughness implementation and convective heat transfer coefficient computation toward ice accretion simulation

In the simulation of icing, the calculation of the convective heat transfer coefficient for rough surfaces has significant importance. Our study investigates the extension of the γ-Reθt transition model for rough icing surfaces and aims to provide an alternative to the integral boundary layer method for facilitating the calculation of the heat transfer coefficient in three-dimensional icing simulations. Initially, the performance of the γ-Reθt transition model in predicting the transition characteristics and heat transfer for smooth surfaces is examined. Subsequently, to compensate for the deficiency of the γ-Reθt transition model in predicting the premature transition caused by roughness, an additional model equation Ar is introduced to simulate the influence of the surface roughness. Three test cases, including a clean cylinder, an iced cylinder, and a swept wing, are used to validate the applicability of the γ-Reθt-Ar transition model for heat transfer computations on rough surfaces. The results demonstrate that the γ-Reθt-Ar model is superior to other turbulence models in computing convective heat transfer for two-dimensional and three-dimensional icing rough surfaces and shows its potential for ice accretion simulation.

Puckered Zigzag Monolayer Ice: Does a Confined Flat Four-Coordinated Monolayer Ice Always Have a Corresponding Puckered Phase?

We note that a flat, four-coordinated monolayer ice under confinement always has a corresponding puckered phase. Recently, a monolayer ice consisting of an array of zigzag water chains (ZZMI) predicted by first-principles calculations of water under confinement is a flat four-coordinated monolayer ice. Herein, to investigate whether puckered ZZMI exists stably, we perform molecular dynamics simulations of two-dimensional (2D) ice formation for water constrained in graphene nanocapillaries. We find a novel monolayer ice structure that can be viewed as the ZZMI puckered along the direction perpendicular to the zigzag chain (pZZMI). Unlike ZZMI that does not satisfy the ice rule, each water molecule in pZZMI can form four hydrogen bonds (HBs) via forming two stable intersublayer HBs and two intrasublayer HBs. This work provides a fresh perspective on 2D confined ice, highlighting the intrinsic connections between 2D confined ices.

Theoretical Prediction of the Anti-Icing Activity of Two-Dimensional Ice I.

Two-dimensional (2D) ice I is atomic-level ice that is composed of two interlocked atomic layers saturated with hydrogen bonds. It has recently been experimentally observed, but its properties have yet to be clarified. Accordingly, we theoretically studied the hydrophobic properties of 2D ice I. On the contrary, a simulation of a hydrogen fluoride molecule on a 2D ice surface manifested that it destroyed the 2D ice structure and connected new hydrogen bonds with water molecules. Investigations of the interfacial effect between 2D and three-dimensional (3D) ice films indicated that the network structure of 2D ice was not destroyed by a 3D ice surface, as the former was saturated with hydrogen bonds. However, the surface of 3D ice reorganized to form as many hydrogen bonds as possible. Thus, the 2D ice film was hydrophobic and inhibited the growth of 3D ice. This shows that if 2D ice can be produced on an industrial scale, it can be used as an anti-3D-icing agent under low temperatures.

Modeling of icing on a planar surface due to Sea spray and blowing Snow

Two main extensions of the time-dependent two-dimensional sea spray icing model ICEMOD2.1 (Horjen, 2015) is presented. The first extension, called ICEMOD2.2, is the inclusion of heat conduction through the accreted ice. This quantity has been calculated from the temperature distribution in the ice which varies with time. The second extension, called ICEMOD2.3, is the combined effect of sea spray and snow. No other icing models have ever before included this effect. The new models have been applied for some reported icing cases on the nearly vertical wall of the bulkhead of the large Norwegian coast guard vessel KV Nordkapp from 1983 to 1997. Local wind field above the boundary layer, heat loss by forced convection and wind stress have been approximated by using analytical formulas for the wind field around a single plate. The physics of instantaneous freezing of a fraction of supercooled droplets upon impact has been discussed and implemented in the model.The most severe icing conditions in Norwegian Arctic water often occur during polar lows, which are often followed up with heavy snow fall. Due to the inclusion of snow the thermal conductivity of the ice accretion is modified. Using an electrical analogy it is shown that this parameter decreases during snow conditions. Overall ice salinity and density are also affected by the snow.

Ice Velocity Variations of the Cook Ice Shelf, East Antarctica, from 2017 to 2022 from Sentinel-1 SAR Time-Series Offset Tracking

The Cook Glacier drains a significant portion of the Wilkes Subglacial Basin, the largest subglacial basin in East Antarctica—which feeds the Cook Ice Shelf. The ice velocity of the Cook Ice Shelf needs to be monitored precisely and accurately, as it plays a critical role in determining the ice discharge from the Wilkes Subglacial Basin. In this study, we measured the annual ice velocities of the Cook Ice Shelf using the offset tracking technique on Sentinel-1 synthetic aperture radar images obtained from 2017 to 2022. Time-series offsets in the range and azimuth directions were determined from the offset tracking pairs with a temporal baseline of 36 days obtained from January to December of each year. Statistical evaluations of the spatiotemporal variations of the time-series offsets effectively eliminated the erroneous offsets in the original offset fields; the remaining offsets were then used to produce two-dimensional annual ice velocities. The direction of the ice flow of the Cook Ice Shelf was almost constant during the period 2017–2022, and the variations in the magnitude of annual ice velocities were investigated. The annual ice velocities of the Cook East Ice Shelf (CEIS) stayed constant and showed a gradual increase from the grounding line to the ice front, except in the western part. Ice velocities of the western part of the CEIS have not changed much at the grounding line during the 6-year period, while in the dynamic shelf ice zone, ice velocities accelerated by up to 22% because of the development of numerous crevasses and fractures. The ice velocities of the Cook West Ice Shelf (CWIS) were about two times higher than those of the CEIS and tended to increase rapidly from the grounding line to the ice front. The annual ice velocities at the grounding line of CWIS increased rapidly from 1330 to 1450 m/a over 6 years, with 70% of this acceleration observed after 2021. This was attributed to a reduction in the ice shelf volume because of the evolution of surface crevasses and rifts, leading to a decrease in the ice shelf’s buttressing potential. In particular, the loss of a portion of the dynamic shelf ice zone due to a series of ice front collapses in February 2022 likely caused the rapid speed-up of the ice shelf. The results of this study indicate that the buttressing potential of the CWIS and the western part of the CEIS has been significantly reduced, which could mean serious instability of the marine ice sheet in this region.

Exact results for the residual entropy of ice hexagonal monolayer.

Since the problem of the residual entropy of square ice was exactly solved, exact solutions for two-dimensional realistic ice models have been of interest. In this work, we study the exact residual entropy of ice hexagonal monolayer in two cases. In the case that the external electric field along the z-axis exists, we map the hydrogen configurations into the spin configurations of the Ising model on the kagome lattice. By taking the low temperature limit of the Ising model, we derive the exact residual entropy, which agrees with the result determined previously from the dimer model on the honeycomb lattice. In another case that the ice hexagonal monolayer is under the periodic boundary conditions in the cubic ice lattice, the residual entropy has not been studied exactly. For this case, we employ the six-vertex model on the square lattice to represent the hydrogen configurations obeying the ice rules. The exact residual entropy is obtained from the solution of the equivalent six-vertex model. Our work provides more examples of the exactly soluble two-dimensional statistical models.

Low-Dimensional and Confined Ice

Owing to its unique structure, morphology, and crystal quality, low-dimensional (L-D) ice has attracted increasing attention in recent years. With a size (at least in one dimension) between that of a single water molecule and a snowflake, L-D ice does not only appear as an intermediate state during the dimensional change but can also manifest extraordinary characteristics, from its molecular structures to its physical properties, which offer exciting opportunities for a better understanding and utilization of ice. In this article, we start with a brief introduction to the crystal growth, structure, and typical characterization techniques of ice and then review recent progress in the study of crystal growth, molecular structures, phase morphologies, and physical properties of zero-, one-, and two-dimensional (0-, 1-, and 2D) ice. Extraordinary behaviors of ice in low dimensions and extreme conditions are highlighted. Finally, the future outlook for the physical study and technological applications of L-D ice is briefly discussed.