Wrinkling composite sheets.

We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the beam equation describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain lengths of the confined sheet; we explain this degeneracy using an asymptotic analysis valid for long sheets, and show that it corresponds to the switching of the sheet between symmetric and antisymmetric buckling modes. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets concentrate the bending energy towards the soft regions of the sheet.

Stochastic paleoclimatology: Modeling the EPICA ice core climate records.

We analyze and model the stochastic behavior of paleoclimate time series and assess the implications for the coupling of climate variables during the Pleistocene glacial cycles. We examine 800 kiloyears of carbon dioxide, methane, nitrous oxide, and temperature proxy data from the European Project for Ice Coring in Antarctica (EPICA) Dome-C ice core, which are characterized by 100 ky glacial cycles overlain by fluctuations across a wide range of timescales. We quantify this behavior through multifractal time-weighted detrended fluctuation analysis, which distinguishes near-red-noise and white-noise behavior below and above the 100 ky glacial cycle, respectively, in all records. This allows us to model each time series as a one-dimensional periodic nonautonomous stochastic dynamical system, and assess the stability of physical processes and the fidelity of model-simulated time series. We extend this approach to a four-variable model with intervariable coupling terms, which we interpret in terms of possible interrelationships among the four time series. Within the framework of our coupling coefficients, we find that carbon dioxide and temperature act to stabilize each other and methane and nitrous oxide, whereas the latter two destabilize each other and carbon dioxide and temperature. We also compute the response function for each pair of variables to assess the model performance by comparison to the data and confirm the model predictions regarding stability amongst variables. Taken together, our results are consistent with glacial pacing dominated by carbon dioxide and temperature that is modulated by terrestrial biosphere feedbacks associated with methane and nitrous oxide emissions.

Premelting controlled active matter in ice.

Self-propelled particles can undergo complex dynamics due to a range of bulk and surface interactions. When a particle is embedded in a host solid near its bulk melting temperature, the latter may melt at the surface of the former in a process known as interfacial premelting. The thickness of the melt film depends on the temperature, impurities, material properties and geometry. A temperature gradient is accompanied by a thermomolecular pressure gradient that drives the interfacial liquid from high to low temperatures and hence the particle from low to high temperatures, in a process called thermal regelation. When the host material is ice and the embedded particle is a biological entity, one has a particularly different form of active matter, which addresses interplay between a wide range of problems, from extremophiles of both terrestrial and exobiological relevance to ecological dynamics in Earth's cryosphere. Of basic importance in all such settings is the combined influence of biological activity and thermal regelation in controlling the redistribution of bioparticles. Therefore, we recast this class of regelation phenomena in the stochastic framework of active Ornstein-Uhlenbeck dynamics and make predictions relevant to this and related problems of interest in biological and geophysical problems. We examine how thermal regelation compromises paleoclimate studies in the context of ice core dating and we find that the activity influences particle dynamics during thermal regelation by enhancing the effective diffusion coefficient. Therefore, accurate dating relies on a quantitative treatment of both effects.

Analytical solution of stochastic resonance in the nonadiabatic regime.

We generalize stochastic resonance to the nonadiabatic limit by treating the double-well potential using two quadratic potentials. We use a singular perturbation method to determine an approximate analytical solution for the probability density function that asymptotically connects local solutions in boundary layers near the two minima with those in the region of the maximum that separates them. The validity of the analytical solution is confirmed numerically. Free from the constraints of the adiabatic limit, the approach allows us to predict the escape rate from one stable basin to another for systems experiencing a more complex periodic forcing.

Predicting critical transitions in multiscale dynamical systems using reservoir computing.

We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as the limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. We discuss the mathematical and broader implications of our results.

Impurity effects in thermal regelation.

When a particle is placed in a material with a lower bulk melting temperature, intermolecular forces can lead to the existence of a "premelted" liquid film of the lower melting temperature material. Despite the system being below the melting temperatures of both solids, the liquid film is a consequence of thermodynamic equilibrium, controlled by intermolecular, ionic and other interactions. An imposed temperature gradient drives the translation of the particle by a process of melting and refreezing known as "thermal regelation". We calculate the rate of regelation of spherical particles surrounded by premelted films that contain ionic impurities. The impurities enhance the rate of motion thereby influencing the dynamics of single particles and distributions of particles, which we describe in addition to the consequences in natural and technological settings.

Surface phase transitions in ice: from fundamental interactions to applications.

Interfaces divide all phases of matter and yet in most practical settings it is tempting to ignore their energies and the associated implications. There are many reasons for this, not the least of which is the introduction of a new pair of canonically conjugate variables-interfacial energy and its counterpart the surface area. A key set of questions surrounding the treatment of multiphase flows concerns how and when we must account for such effects. I begin this discussion with an abbreviated review of the basic theory of lower-dimensional phase transitions and describe a range of situations in which the bulk behaviour of a two-phase (and in some cases two-component) system is dominated by surface effects. Then I discuss a number of settings in which the bulk and surface behaviour can interact on equal footing. These can include the dynamic and thermodynamic behaviour of floating sea ice, the freezing and drying of colloidal suspensions (such as soil) and the mechanisms of protoplanetesimal formation by inter-particle collisions in accretion discs. This article is part of the theme issue 'The physics and chemistry of ice: scaffolding across scales, from the viability of life to the formation of planets'.

Intrinsic Pink-Noise Multidecadal Global Climate Dynamics Mode.

Understanding multidecadal variability is an essential goal of climate dynamics. For example, the recent phenomenon referred to as the "global warming hiatus" may reflect a coupling to an intrinsic, preindustrial, multidecadal variability process. Here, using a multifractal time-series method, we demonstrate that 42 data sets of 79 proxies with global coverage exhibit pink-noise characteristics on multidecadal timescales. To quantify the persistence of this behavior, we examine high-resolution ice core and speleothem data to find pink noise in both pre- and postindustrial periods. We examine the spatial structure with an empirical orthogonal function analysis of the monthly averaged surface temperature from 1901 to 2012. The first mode clearly shows the distribution of ocean heat flux sinks located in the eastern Pacific and the Southern Ocean and has pink-noise characteristics on a multidecadal timescale. We hypothesize that this pink-noise multidecadal spatial mode may resonate with externally driven greenhouse gas forcing, driving large-scale climate processes.

Statistical Mechanics and the Climatology of the Arctic Sea Ice Thickness Distribution.

We study the seasonal changes in the thickness distribution of Arctic sea ice, g(h), under climate forcing. Our analytical and numerical approach is based on a Fokker-Planck equation for g(h) (Toppaladoddi and Wettlaufer in Phys Rev Lett 115(14):148501, 2015), in which the thermodynamic growth rates are determined using observed climatology. In particular, the Fokker-Planck equation is coupled to the observationally consistent thermodynamic model of Eisenman and Wettlaufer (Proc Natl Acad Sci USA 106:28-32, 2009). We find that due to the combined effects of thermodynamics and mechanics, g(h) spreads during winter and contracts during summer. This behavior is in agreement with recent satellite observations from CryoSat-2 (Kwok and Cunningham in Philos Trans R Soc A 373(2045):20140157, 2015). Because g(h) is a probability density function, we quantify all of the key moments (e.g., mean thickness, fraction of thin/thick ice, mean albedo, relaxation time scales) as greenhouse-gas radiative forcing, Δ F 0 , increases. The mean ice thickness decays exponentially with Δ F 0 , but much slower than do solely thermodynamic models. This exhibits the crucial role that ice mechanics plays in maintaining the ice cover, by redistributing thin ice to thick ice-far more rapidly than can thermal growth alone.

Theory of the Sea Ice Thickness Distribution.

We use concepts from statistical physics to transform the original evolution equation for the sea ice thickness distribution g(h) from Thorndike et al. into a Fokker-Planck-like conservation law. The steady solution is g(h)=N(q)h(q)e(-h/H), where q and H are expressible in terms of moments over the transition probabilities between thickness categories. The solution exhibits the functional form used in observational fits and shows that for h≪1, g(h) is controlled by both thermodynamics and mechanics, whereas for h≫1 only mechanics controls g(h). Finally, we derive the underlying Langevin equation governing the dynamics of the ice thickness h, from which we predict the observed g(h). The genericity of our approach provides a framework for studying the geophysical-scale structure of the ice pack using methods of broad relevance in statistical mechanics.

On the nature of the sea ice albedo feedback in simple models.

We examine the nature of the ice-albedo feedback in a long-standing approach used in the dynamic-thermodynamic modeling of sea ice. The central issue examined is how the evolution of the ice area is treated when modeling a partial ice cover using a two-category-thickness scheme; thin sea ice and open water in one category and "thick" sea ice in the second. The problem with the scheme is that the area evolution is handled in a manner that violates the basic rules of calculus, which leads to a neglected area evolution term that is equivalent to neglecting a leading-order latent heat flux. We demonstrate the consequences by constructing energy balance models with a fractional ice cover and studying them under the influence of increased radiative forcing. It is shown that the neglected flux is particularly important in a decaying ice cover approaching the transitions to seasonal or ice-free conditions. Clearly, a mishandling of the evolution of the ice area has leading-order effects on the ice-albedo feedback. Accordingly, it may be of considerable importance to reexamine the relevant climate model schemes and to begin the process of converting them to fully resolve the sea ice thickness distribution in a manner such as remapping, which does not in principle suffer from the pathology we describe.

Grain boundary melting in ice.

We describe an optical scattering study of grain boundary premelting in water ice. Ubiquitous long ranged attractive polarization forces act to suppress grain boundary melting whereas repulsive forces originating in screened Coulomb interactions and classical colligative effects enhance it. The liquid enhancing effects can be manipulated by adding dopant ions to the system. For all measured grain boundaries this leads to increasing premelted film thickness with increasing electrolyte concentration. Although we understand that the interfacial surface charge densities q(s) and solute concentrations can potentially dominate the film thickness, we cannot directly measure them within a given grain boundary. Therefore, as a framework for interpreting the data we consider two appropriate q(s) dependent limits; one is dominated by the colligative effect and other is dominated by electrostatic interactions.

Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses.

Droplets deform soft substrates near their contact lines. Using confocal microscopy, we measure the deformation of silicone gel substrates due to glycerol and fluorinated-oil droplets for a range of droplet radii and substrate thicknesses. For all droplets, the substrate deformation takes a universal shape close to the contact line that depends on liquid composition, but is independent of droplet size and substrate thickness. This shape is determined by a balance of interfacial tensions at the contact line and provides a novel method for direct determination of the surface stresses of soft substrates. Moreover, we measure the change in contact angle with droplet radius and show that Young's law fails for small droplets when their radii approach an elastocapillary length scale. For larger droplets the macroscopic contact angle is constant, consistent with Young's law.

New insights into ice growth and melting modifications by antifreeze proteins.

Antifreeze proteins (AFPs) evolved in many organisms, allowing them to survive in cold climates by controlling ice crystal growth. The specific interactions of AFPs with ice determine their potential applications in agriculture, food preservation and medicine. AFPs control the shapes of ice crystals in a manner characteristic of the particular AFP type. Moderately active AFPs cause the formation of elongated bipyramidal crystals, often with seemingly defined facets, while hyperactive AFPs produce more varied crystal shapes. These different morphologies are generally considered to be growth shapes. In a series of bright light and fluorescent microscopy observations of ice crystals in solutions containing different AFPs, we show that crystal shaping also occurs during melting. In particular, the characteristic ice shapes observed in solutions of most hyperactive AFPs are formed during melting. We relate these findings to the affinities of the hyperactive AFPs for the basal plane of ice. Our results demonstrate the relation between basal plane affinity and hyperactivity and show a clear difference in the ice-shaping mechanisms of most moderate and hyperactive AFPs. This study provides key aspects associated with the identification of hyperactive AFPs.

Density functional theory for Baxter's sticky hard spheres in confinement.

It has recently been shown that a free energy for Baxter's sticky hard-sphere fluid is uniquely defined within the framework of fundamental measure theory (FMT) for the inhomogeneous hard-sphere fluid, provided that it obeys scaled-particle theory and the Percus-Yevick (PY) result for the direct correlation function [H. Hansen-Goos and J. S. Wettlaufer, J. Chem. Phys. 134, 014506 (2011)]. Here, combining FMT weighted densities with a new vectorial weighted density, we regularize the divergences of the associated strongly confined limit. The free energy that emerges is exact in the zero-dimensional limit, leaves the underlying equation of state unaffected, and yields a direct correlation function distinct from that of PY. Comparison with simulation data for both the bulk pair-correlation function and the density profiles in confinement shows that the new theory is significantly more accurate than the PY-based results.

Ice-lens formation and geometrical supercooling in soils and other colloidal materials.

We present a physically intuitive model of ice-lens formation and growth during the freezing of soils and other dense, particulate suspensions. Motivated by experimental evidence, we consider the growth of an ice-filled crack in a freezing soil. At low temperatures, ice in the crack exerts large pressures on the crack walls that will eventually cause the crack to split open. We show that the crack will then propagate across the soil to form a new lens. The process is controlled by two factors: the cohesion of the soil and the geometrical supercooling of the water in the soil, a new concept introduced to measure the energy available to form a new ice lens. When the supercooling exceeds a critical amount (proportional to the cohesive strength of the soil) a new ice lens forms. This condition for ice-lens formation and growth does not appeal to any ad hoc, empirical assumptions, and explains how periodic ice lenses can form with or without the presence of a frozen fringe. The proposed mechanism is in good agreement with experiments, in particular explaining ice-lens pattern formation and surges in heave rate associated with the growth of new lenses. Importantly for systems with no frozen fringe, ice-lens formation and frost heave can be predicted given only the unfrozen properties of the soil. We use our theory to estimate ice-lens growth temperatures obtaining quantitative agreement with the limited experimental data that are currently available. Finally we suggest experiments that might be performed in order to verify this theory in more detail. The theory is generalizable to complex natural-soil scenarios and should therefore be useful in the prediction of macroscopic frost-heave rates.

Dynamics of colloidal particles in ice.

We use x-ray photon correlation spectroscopy (XPCS) to probe the dynamics of colloidal particles in polycrystalline ice. During freezing, the dendritic ice morphology and rejection of particles from the ice created regions of high particle density, where some of the colloids were forced into contact and formed disordered aggregates. The particles in these high density regions underwent ballistic motion, with a characteristic velocity that increased with temperature. This ballistic motion is coupled with both stretched and compressed exponential decays of the intensity autocorrelation function. We suggest that this behavior could result from ice grain boundary migration.

Particle-scale structure in frozen colloidal suspensions from small-angle x-ray scattering.

During directional solidification of the solvent in a colloidal suspension, the colloidal particles segregate from the growing solid, forming high-particle-density regions with structure on a hierarchy of length scales ranging from that of the particle-scale packing to the large-scale spacing between these regions. Previous work has concentrated mostly on the medium- to large-length scale structure, as it is the most accessible and thought to be more technologically relevant. However, the packing of the colloids at the particle scale is an important component not only in theoretical descriptions of the segregation process, but also to the utility of freeze-cast materials for new applications. Here we present the results of experiments in which we investigated this structure across a wide range of length scales using a combination of small-angle x-ray scattering and direct optical imaging. As expected, during freezing the particles were concentrated into regions between ice dendrites forming a microscopic pattern of high- and low-particle-density regions. X-ray scattering indicates that the particles in the high-density regions were so closely packed as to be touching. However, the arrangement of the particles does not conform to that predicted by standard interparticle pair potentials, suggesting that the particle packing induced by freezing differs from that formed during equilibrium densification processes.