Ice Rules Research Articles

Is Pauling’s Hypothesis Violated in the Hexagonal Modification of Ice?

The effect of long-range dipole interaction on the fulfilment of Pauling’s hypothesis concerning the degeneracy of proton configurations satisfying the ice rules is studied for the case of hexagonal ice. It is shown that the ice rules imply a significant reduction of the contributions to the ground state energy from the next-nearest neighbors. The ground state energy of proton subsystem as a function of quasimomentum ceases to change if more than a hundred primitive unit cells in each direction are taken into account. The final amplitude of the energy changes are about 70 K. The analysis reveals the proton ordering in hexagonal ice with the simple unit cell of the oxygen sublattice is impossible. However, that does not exclude proton ordering with a more complex cell caused by distortion of the hexagonal oxygen lattice.

Interplay of defect doping and Bernal-Fowler rules: A simulation study of the dynamics on ice lattices

Protonic defects on ice lattices induced by doping with acids such as HCl and HF or bases such as KOH can facilitate order-disorder transitions. In laboratory experiments KOH doping is efficient in promoting the ordering transition from hexagonal ice I to ice XI, but it is ineffective for other known ice phases, for which HCl can trigger hydrogen ordering. Aiming at understanding these differences, random-walk simulations of the defect diffusion are performed on two- and three-dimensional ice lattices under the constraints imposed by the Bernal-Fowler ice rules. Effective defect diffusion coefficients are calculated for a range of dopants, concentrations, and ice phases. The interaction of different defects, incorporated by different dopants, is investigated to clarify the particular motion-enhancing role played by complementary defect pairs.

Water-Ice Analogues of Polycyclic Aromatic Hydrocarbons: Water Nanoclusters on Cu(111).

Water has an incredible ability to form a rich variety of structures, with 16 bulk ice phases identified, for example, as well as numerous distinct structures for water at interfaces or under confinement. Many of these structures are built from hexagonal motifs of water molecules, and indeed, for water on metal surfaces, individual hexamers of just six water molecules have been observed. Here, we report the results of low-temperature scanning tunneling microscopy experiments and density functional theory calculations which reveal a host of new structures for water–ice nanoclusters when adsorbed on an atomically flat Cu surface. The H-bonding networks within the nanoclusters resemble the resonance structures of polycyclic aromatic hydrocarbons, and water–ice analogues of inene, naphthalene, phenalene, anthracene, phenanthrene, and triphenylene have been observed. The specific structures identified and the H-bonding patterns within them reveal new insight about water on metals that allows us to refine the so-called “2D ice rules”, which have so far proved useful in understanding water–ice structures at solid surfaces.

Emergent geometric frustration of artificial magnetic skyrmion crystals

Magnetic skyrmions have been receiving growing attention as potential information storage and magnetic logic devices since an increasing number of materials have been identified that support skyrmion phases. Explorations of artificial frustrated systems have led to new insights into controlling and engineering new emergent frustration phenomena in frustrated and disordered systems. Here, we propose a skyrmion spin ice, giving a unifying framework for the study of geometric frustration of skyrmion crystals in a non-frustrated artificial geometrical lattice as a consequence of the structural confinement of skyrmions in magnetic potential wells. The emergent ice rules from the geometrically frustrated skyrmion crystals highlight a novel phenomenon in this skyrmion system: emergent geometrical frustration. We demonstrate how skyrmion crystal topology transitions between a non-frustrated periodic configuration and a frustrated ice-like ordering can also be realized reversibly. The proposed artificial frustrated skyrmion systems can be annealed into different ice phases with an applied current induced spin-transfer torque, including a long range ordered ice rule obeying ground state, as-relaxed random state, biased state and monopole state. The spin-torque reconfigurability of the artificial skyrmion ice states, difficult to achieve in other artificial spin ice systems, is compatible with standard spintronic device fabrication technology, which makes the semiconductor industrial integration straightforward.

Nonlinear geometric scaling of coercivity in a three-dimensional nanoscale analog of spin ice

Magnetization hysteresis loops of a three-dimensional nanoscale analog of spin ice based on the nickel inverse opal-like structure (IOLS) have been studied at room temperature. The samples are produced by filling nickel into the voids of artificial opal-like films. The spin ice behavior is induced by tetrahedral elements within the IOLS, which have the same arrangement of magnetic moments as a spin ice. The thickness of the films vary from a two-dimensional, i.e., single-layered, antidot array to a three-dimensional, i.e., multilayered, structure. The coercive force, the saturation, and the irreversibility field have been measured in dependence of the thickness of the IOLS for in-plane and out-of-plane applied fields. The irreversibility and saturation fields change abruptly from the antidot array to the three-dimensional IOLS and remain constant upon further increase of the number of layers $n$. The coercive force ${H}_{c}$ seems to increase logarithmically with increasing $n$ as ${H}_{c}={H}_{c0}+\ensuremath{\alpha}ln(n+1)$. The logarithmic law implies the avalanchelike remagnetization of anisotropic structural elements connecting tetrahedral and cubic nodes in the IOLS. We conclude that the ``ice rule'' is the base of mechanism regulating this process.

Not obeying the rules

The physical properties of ice are governed by its tetrahedral network of hydrogen bonds and the ice rules that determine the distribution of the protons. Deviations from the tetrahedral structure and violations of these rules can lead to surprising phenomena, such as the ferroelectric state now reported for thin films of epitaxial ice.

Classical and quantum theories of proton disorder in hexagonal water ice

It has been known since the pioneering work of Bernal, Fowler and Pauling that common, hexagonal (Ih) water ice is the archetype of a frustrated material : a proton-bonded network in which protons satisfy strong local constraints - the "ice rules" - but do not order. While this proton disorder is well established, there is now a growing body of evidence that quantum effects may also have a role to play in the physics of ice at low temperatures. In this Article we use a combination of numerical and analytic techniques to explore the nature of proton correlations in both classical and quantum models of ice Ih. In the case of classical ice Ih, we find that the ice rules have two, distinct, consequences for scattering experiments - singular "pinch points", reflecting a zero-divergence condition on the uniform polarization of the crystal, and broad, asymmetric features, coming from its staggered polarisation. In the case of the quantum model, we find that the collective quantum tunnelling of groups of protons can convert states obeying the ice rules into a quantum liquid, whose excitations are birefringent, emergent photons. We make explicit predictions for scattering experiments on both classical and quantum ice Ih, and show how the quantum theory can explain the "wings" of incoherent inelastic scattering observed in recent neutron scattering experiments [Bove et al., Phys. Rev. Lett. 103, 165901 (2009)]. These results raise the intriguing possibility that the protons in ice Ih could form a quantum liquid at low temperatures, in which protons are not merely disordered, but continually fluctuate between different configurations obeying the ice rules.

Structural and configurational properties of nanoconfined monolayer ice from first principles.

Understanding the structural tendencies of nanoconfined water is of great interest for nanoscience and biology, where nano/micro-sized objects may be separated by very few layers of water. Here we investigate the properties of ice confined to a quasi-2D monolayer by a featureless, chemically neutral potential, in order to characterize its intrinsic behaviour. We use density-functional theory simulations with a non-local van der Waals density functional. An ab initio random structure search reveals all the energetically competitive monolayer configurations to belong to only two of the previously-identified families, characterized by a square or honeycomb hydrogen-bonding network, respectively. We discuss the modified ice rules needed for each network, and propose a simple point dipole 2D lattice model that successfully explains the energetics of the square configurations. All identified stable phases for both networks are found to be non-polar (but with a topologically non-trivial texture for the square) and, hence, non-ferroelectric, in contrast to previous predictions from a five-site empirical force-field model. Our results are in good agreement with very recently reported experimental observations.

Pressure-induced localisation of the hydrogen-bond network in KOH-VI.

Using a combination of ab initio crystal structure prediction and neutron diffraction techniques, we have solved the full structure of KOH-VI at 7 GPa. Rather than being orthorhombic and proton-ordered as had previously be proposed, we find that this high-pressure phase of potassium hydroxide is tetragonal (space group I4/mmm) and proton disordered. It has an unusual hydrogen bond topology, where the hydroxyl groups form isolated hydrogen-bonded square planar (OH)4 units. This structure is stable above 6.5 GPa and, despite being macroscopically proton-disordered, local ice rules enforce microscopic order of the hydrogen bonds. We suggest the use of this novel type of structure to study concerted proton tunneling in the solid state, while the topology of the hydrogen bond network could conceivably be exploited in data storage applications based solely on the manipulations of hydrogen bonds. The unusual localisation of the hydrogen bond network under applied pressure is found to be favored by a more compact packing of the constituents in a distorted cesium chloride structure.

Thermodynamic Stability of Ice II and Its Hydrogen-Disordered Counterpart: Role of Zero-Point Energy

We investigate why no hydrogen-disordered form of ice II has been found in nature despite the fact that most of hydrogen-ordered ices have hydrogen-disordered counterparts. The thermodynamic stability of a set of hydrogen-ordered ice II variants relative to ice II is evaluated theoretically. It is found that ice II is more stable than the disordered variants so generated as to satisfy the simple ice rule due to the lower zero-point energy as well as the pair interaction energy. The residual entropy of the disordered ice II phase gradually compensates the unfavorable free energy with increasing temperature. The crossover, however, occurs at a high temperature well above the melting point of ice III. Consequently, the hydrogen-disordered phase does not exist in nature. The thermodynamic stability of partially hydrogen-disordered ices is also scrutinized by examining the free-energy components of several variants obtained by systematic inversion of OH directions in ice II. The potential energy of one variant is lower than that of the ice II structure, but its Gibbs free energy is slightly higher than that of ice II due to the zero-point energy. The slight difference in the thermodynamic stability leaves the possibility of the partial hydrogen-disorder in real ice II.

Similarity of the dielectric relaxation processes and transport characteristics in water and ice

The electrical characteristics of ice and water in wide temperature and frequency ranges are analyzed and compared. It is found that the experimental data agree well with the conjecture that the mechanisms of electrical relaxation in ice and water are similar to each other. In particular, it is shown that the theory of electrical relaxation developed for ice gives a quantitative description of the same data for water under the assumption that the activation energy of defects (violations of the ice rule) is lower in water. The reasons for the similarity of electrical relaxation mechanisms and possible applications of such similarity are discussed.

Novel Magnetic Orders and Ice Phases in Frustrated Kondo-Lattice Models

We review recent theoretical progress in our understanding of electron-driven novel magnetic phases on frustrated lattices. Our specific focus is on Kondo-lattice or double-exchange models assuming finite magnetic moments localized at the lattice sites. A salient feature of systems with SU(2) symmetric local moments is the emergence of noncoplanar magnetic ordering driven by the conduction electrons. The complex spin textures then endow the electrons a nontrivial Berry phase, often giving rise to a topologically nontrivial electronic state. The second part of the review is devoted to the discussion of metallic spin ice systems, which are essentially frustrated Ising magnets with local spin ordering governed by the so-called ice rules. These rules are similar to those that describe proton configurations in solid water ice, hence the name "spin ice". The nontrivial spin correlations in the ice phase give rise to unusual electron transport properties in metallic spin-ice systems.

Ice rule for a ferromagnetic nanosite network on the face-centered cubic lattice

The magnetic properties of an inverse opal-like cobalt-based structure having the symmetry of the face-centered cubic lattice are studied. The magnetization reversal of the structure in a magnetic field applied along the [\(1\overline 1 0\)] axis is described using a phenomenological model, which uses the ice rule for the local magnetization of nanostructure elements. This description predicts the absence of a long-range magnetic order in two directions that are normal to the magnetic field. The magnetic structure is analyzed by smallangle neutron diffraction. Neutron diffraction patterns are measured in an external magnetic field varying from −1.2 to 1.2 T and applied along the crystallographic [\(1\overline 1 0\)] direction. A magnetic contribution to the neutron scattering intensity is extracted. It is shown that this contribution is close to zero over the entire applied magnetic field range for all scattering planes that are normal to the field, which agrees well with the proposed model.

Highly confined water: two-dimensional ice, amorphous ice, and clathrate hydrates.

Understanding phase behavior of highly confined water, ice, amorphous ice, and clathrate hydrates (or gas hydrates), not only enriches our view of phase transitions and structures of quasi-two-dimensional (Q2D) solids not seen in the bulk phases but also has important implications for diverse phenomena at the intersection between physical chemistry, cell biology, chemical engineering, and nanoscience. Relevant examples include, among others, boundary lubrication in nanofluidic and lab-on-a-chip devices, synthesis of antifreeze proteins for ice-growth inhibition, rapid cooling of biological suspensions or quenching emulsified water under high pressure, and storage of H2 and CO2 in gas hydrates. Classical molecular simulation (MD) is an indispensable tool to explore states and properties of highly confined water and ice. It also has the advantage of precisely monitoring the time and spatial domains in the sub-picosecond and sub-nanometer scales, which are difficult to control in laboratory experiments, and yet allows relatively long simulation at the 10(2) ns time scale that is impractical with ab initio molecular dynamics simulations. In this Account, we present an overview of our MD simulation studies of the structures and phase behaviors of highly confined water, ice, amorphous ice, and clathrate, in slit graphene nanopores. We survey six crystalline phases of monolayer (ML) ice revealed from MD simulations, including one low-density, one mid-density, and four high-density ML ices. We show additional supporting evidence on the structural stabilities of the four high-density ML ices in the vacuum (without the graphene confinement), for the first time, through quantum density-functional theory optimization of their free-standing structures at zero temperature. In addition, we summarize various low-density, high-density, and very-high-density Q2D bilayer (BL) ice and amorphous ice structures revealed from MD simulations. These simulations reinforce the notion that the nanoscale confinement not only can disrupt the hydrogen bonding network in bulk water but also can allow satisfaction of the ice rule for low-density and high-density Q2D crystalline structures. Highly confined water can serve as a generic model system for understanding a variety of Q2D materials science phenomena, for example, liquid-solid, solid-solid, solid-amorphous, and amorphous-amorphous transitions in real time, as well as the Ostwald staging during these transitions. Our simulations also bring new molecular insights into the formation of gas hydrate from a gas and water mixture at low temperature.

Emergent ice rule and magnetic charge screening from vertex frustration in artificial spin ice

The shakti lattice describes a new type of frustration not found in naturally occurring materials. Fabrication of the first artificial spin-ice array displaying shakti dynamics confirms the locally ordered, globally degenerate nature of these exotic lattice structures.

Gauge field and the confinement-deconfinement transition in hydrogen-bonded ferroelectrics.

Quantum melting of a ferroelectric moment in the frustrated hydrogen-bonded system with the "ice rule" is studied theoretically by using quantum Monte Carlo simulation. The large number of nearly degenerate configurations are described as the gauge degrees of freedom; i.e., the model is mapped to a lattice gauge theory, which shows the confinement-deconfinment transition (CDT). The dipole-dipole interaction J(2), on the other hand, explicitly breaks the gauge symmetry leading to the ferroelectric transition at finite temperature T. It is found that the crossover from the FT to CDT manifests itself in the reduced correlation length of the polarization ξ(FT)∼Δ(K-K(c))(-ν), with Δ∝√J(2) while K(c) and ν remains finite in the limit J(2)→0. In contrast, the Curie-Weiss-like law for the susceptibility χ and the spontaneous polarization behaves smoothly and the length scale ξ(CDT), related to the molecular symmetry and volume for CDT, does not reduce in this limit.

Configurational entropy of hydrogen-disordered ice polymorphs.

The configurational entropy of several H-disordered ice polymorphs is calculated by means of a thermodynamic integration along a path between a totally H-disordered state and one fulfilling the Bernal-Fowler ice rules. A Monte Carlo procedure based on a simple energy model is used, so that the employed thermodynamic path drives the system from high temperatures to the low-temperature limit. This method turns out to be precise enough to give reliable values for the configurational entropy sth of different ice phases in the thermodynamic limit (number of molecules N → ∞). The precision of the method is checked for the ice model on a two-dimensional square lattice. Results for the configurational entropy are given for H-disordered arrangements on several polymorphs, including ices Ih, Ic, II, III, IV, V, VI, and XII. The highest and lowest entropy values correspond to ices VI and XII, respectively, with a difference of 3.3% between them. The dependence of the entropy on the ice structures has been rationalized by comparing it with structural parameters of the various polymorphs, such as the mean ring size. A particularly good correlation has been found between the configurational entropy and the connective constant derived from self-avoiding walks on the ice networks.

An unconventional bilayer ice structure on a NaCl(001) film.

Water-solid interactions are of broad importance both in nature and technology. The hexagonal bilayer model based on the Bernal-Fowler-Pauling ice rules has been widely adopted to describe water structuring at interfaces. Using a cryogenic scanning tunnelling microscope, here we report a new type of two-dimensional ice-like bilayer structure built from cyclic water tetramers on an insulating NaCl(001) film, which is completely beyond this conventional bilayer picture. A novel bridging mechanism allows the interconnection of water tetramers to form chains, flakes and eventually a two-dimensional extended ice bilayer containing a regular array of Bjerrum D-type defects. Ab initio density functional theory calculations substantiate this bridging growth mode and reveal a striking proton-disordered ice structure. The formation of the periodic Bjerrum defects with unusually high density may have a crucial role as H donor sites in directing multilayer ice growth and in catalysing heterogeneous chemical reactions on water-coated salt surfaces.

Residual entropy of ices and clathrates from Monte Carlo simulation.

We calculated the residual entropy of ices (Ih, Ic, III, V, VI) and clathrates (I, II, H), assuming the same energy of all configurations satisfying the Bernal-Fowler ice rules. The Metropolis Monte Carlo simulations in the range of temperatures from infinity to a size-dependent threshold were followed by the thermodynamic integration. Convergence of the simulation and the finite-size effects were analyzed using the quasichemical approximation and the Debye-Hückel theory applied to the Bjerrum defects. The leading finite-size error terms, ln N/N, 1/N, and for the two-dimensional square ice model also 1/N(3/2), were used for an extrapolation to the thermodynamic limit. Finally, we discuss the influence of unequal energies of proton configurations.

Hubbard Model on the Pyrochlore Lattice: A 3D Quantum Spin Liquid

We demonstrate that the insulating one-band Hubbard model on the pyrochlore lattice contains, for realistic parameters, an extended quantum spin-liquid phase. This is a three-dimensional spin liquid formed from a highly degenerate manifold of dimer-based states, which is a subset of the classical dimer coverings obeying the ice rules. It possesses spinon excitations, which are both massive and deconfined, and on doping it exhibits spin-charge separation. We discuss the realization of this state in effective S = 1/2 pyrochlore materials with and without spin-orbit coupling.