Shape Relaxation Research Articles

Janus nanoparticles as efficient interface compatibilizer in blends of polylactide and elastomers: Importance of interfacial relaxation on toughening

For blending immiscible polymers, such as in the toughening modification of polylactide (PLA) via blending with rubbery materials, interfacial compatibilization is of great significance while the mechanism, especially the role of interfacial rheology, remains elusive. In this study, styrene-butadiene block copolymer elastomer (SBC) was employed to toughen PLA and a dumbbell-shaped Janus nanoparticle (JNP) consisting of polymethyl methacrylate and polystyrene spheres with equal size (∼80 nm) was used as the compatibilizer. Located at the interface, JNPs exhibited a great compatibilization efficiency in PLA/SBC blends, as demonstrated by the good morphology stabilization against droplet coalescence under static annealing and low shear flow conditions, as well as by the resistance against droplet breakup under high shear flow conditions. Moreover, as revealed from the linear viscoelasticity of JNP compatibilized blends, when JNP loading is more than 2 phr, aside from shape relaxation, an interfacial relaxation dominated by Marangoni stress was observed, indicating the possibility of particle redistribution on droplet surfaces. However, when loading is more than 4 phr, relaxations in the terminal zone no longer exist, implying the possible formation of a particle network on the droplet surface. This is consistent with the mechanical properties. The blend shows the greatest toughness at JNP loading around 3 phr, while the toughness is very poor when JNP loading is either too low or too high. This suggests interfacial relaxation to be crucial to guarantee a good toughening effect of SBC in PLA.

Phase field modeling of hyperelastic material interfaces –Theory, implementation and application to phase transformations

Interface mechanics can significantly govern the evolution of multiple phases on smaller scales, e.g., determining the properties of TWIP- and TRIP-steels, geopolymers or Li-ion batteries. The present contribution is specifically centered around the influence of interface elasticity on mechanically induced phase transformations. A geometrically exact finite element framework is developed for this purpose based on incremental energy minimization. It is set up in three-dimensional space and comprises an Allen-Cahn type phase field formulation based on a Modica-Mortola double-well functional, as well as a Ginzburg-Landau type viscosity formulation. The phase field approximation of the interface deformation gradient is derived by means of homogenization theory. A monolithic solver is employed for the resulting set of non-linear coupled equations. The example of coalescing spheres shows that larger interface energies can amplify and accelerate coalescence. Lower interface energies, in contrast, allow for distinct spreading of the energetically favorable bulk phase. Deformation-dependent interface energies particularly add stress peaks to this process that are localized at the phase transition zone. The relaxation of a stepped interface shape showed that deformation-dependent interface energies favor a plane partition of the bulk phases. Yet, they also delay breakup events compared to simple constant interface energies. This peculiarity can be beneficial for calibration purposes. Strain dependence moreover causes an additional elastic stiffness in the direction of the interface tangent plane and constitutes thus another origin for anisotropy of the overall structure.

A dynamic trap well model of hydrothermal shape-memory effect in amorphous polymers undergoing tailorable shape recovery behaviour

A dynamic trap well model is developed to describe the complex relaxations of functional segments, and explore the working principles behind the hydrothermal coupling effect in shape memory polymers (SMPs). A constitutive relationship among shape fixity strain, shape recovery strain and relaxation time has been formulated to characterize the hydrothermal coupling effect using geometrical parameters (i.e. width and height) of trap wells. Moreover, effects of temperature and solvent absorption on dynamic relaxation behaviours of SMPs have been formulated based on the Flory-Huggins theory and Fokker-Plank probability equation. The trap well model effectively analyzes the shape fixity ratio and shape recovery ratio within ranges of 50–100% and 0–100%, respectively. Finally, an extended Maxwell model is proposed to formulate the dynamic mechanical behaviours of SMPs with hydrothermal shape-memory effect (SME), and the analytical results have been verified using the experimental results reported in literature. A good agreement between the analytical results obtained from the proposed model and the experimental data is present, where the correlation coefficient ( R 2 ) is 95%. The present study firstly introduces the dynamic trap well model for shape memory behaviours and intricate relaxations, and then accurately predicts the dynamic shape recovery of SMP in response to hydrothermal stimulus.

Evolution dynamics of thin liquid structures investigated using a phase-field model.

Liquid structures of thin-films and torus droplets are omnipresent in daily lives. The morphological evolution of liquid structures suspending in another immiscible fluid and sitting on a solid substrate is investigated by using three-dimensional (3D) phase-field (PF) simulations. Here, we address the evolution dynamics by scrutinizing the interplay of surface energy, kinetic energy, and viscous dissipation, which is characterized by Reynolds number Re and Weber number We. We observe special droplet breakup phenomena by varying Re and We. In addition, we gain the essential physical insights into controlling the droplet formation resulting from the morphological evolution of the liquid structures by characterizing the top and side profiles under different circumstances. We find that the shape evolution of the liquid structures is intimately related to the initial shape, Re, We as well as the intrinsic wettability of the substrate. Furthermore, it is revealed that the evolution dynamics are determined by the competition between the coalescence phenomenology and the hydrodynamic instability of the liquid structures. For the coalescence phenomenology, the liquid structure merges onto itself, while the hydrodynamic instability leads to the breakup of the liquid structure. Last but not least, we investigate the influence of wall relaxation on the breakup outcome of torus droplets on substrates with different contact angles. We shed light on how the key parameters including the initial shape, Re, We, wettability, and wall relaxation influence the droplet dynamics and droplet formation. These findings are anticipated to contribute insights into droplet-based systems, potentially impacting areas like ink-jet printing, drug delivery systems, and microfluidic devices, where the interplay of surface energy, kinetic energy, and viscous dissipation plays a crucial role.

A fully decoupled numerical method for Cahn–Hilliard–Navier–Stokes–Darcy equations based on auxiliary variable approaches

A fully decoupled, linearized, and unconditionally stable finite element method is developed to solve the Cahn–Hilliard–Navier–Stokes–Darcy model in the coupled free fluid region and porous medium region. By introducing two auxiliary energy variables, we derive the equivalent system that is consistent with the original system. The energy dissipation law of the proposed equivalent model is proven. To lay a solid foundation, we first present a coupled linearized time-stepping method for the reformulated system, and prove its unconditionally energy stability. In order to further improve the computational efficiency, special treatment for the interface conditions and the artificial compression approach are utilized to decouple the two subdomains and the Navier–Stokes equation. Therefore, with the discretization techniques of two existing auxiliary variable approaches, a fully decoupled and linearized numerical scheme can be developed, under the framework of a semi-implicit semi-explicit scheme for temporal discretization and Galerkin finite element method for spatial discretization. The grad-div stabilization is also employed to further improve the stability of auxiliary variable algorithm. The full discretization obeys the desired energy dissipation law without any temporal restriction. Moreover, the implementation process is discussed, including the adaptive mesh strategy to accurately capture the diffuse interface. Ample numerical experiments are performed to validate the typical features of developed numerical schemes, such as the accuracy, energy stability without restriction for time step size, and adaptive mesh refinement in space. Furthermore, we apply the proposed numerical method to simulate the shape relaxation and the Buoyancy-driven flows, which demonstrate the applicability of the proposed method.

Curvature and shape relaxation in surface-viscous domains.

The mechanics of curved, heterogeneous, surfactant-laden surfaces and interfaces are important to a variety of engineering and biological applications. To date, most models of rheologically complex interfaces have focused on homogeneous systems of planar or fixed curvature. In this study, we investigate a simple, dynamical model of a two-phase surface fluid on a curved interface: a condensed, surface-viscous domain embedded within a surface-inviscid, spherical interface of time-varying radius of curvature. Our aim is to understand how changes in surface curvature generate two-dimensional Stokes flows inside the domain, thereby resisting curvature deformation and distorting the domain shape. We model the surface stress within the domain using the classical Boussinesq-Scriven constitutive equation, simplified for a near-spherical cap undergoing a small-amplitude curvature deformation. We then analyze the frequency-dependent dynamics of the surface stress and curvature within the domain when the pressure difference across the surface is sinusoidally oscillated. We find that the curvature relaxes diffusively, and thus define a Peclet number (Pe) relating the rate of diffusion to the oscillation frequency. At small enough Pe, the surface deforms quasi-statically, whereas at high Pe, the curvature varies sharply within a thin boundary layer adjacent to the domain border. Consequently, the curvature of the domain appears discontinuous from the rest of the surface under rapid oscillation. We then examine the linear stability of the domain shape to small, non-axisymmetric perturbations when the surface is steadily compressed (i.e., the pressure difference across it is increased). While the line tension at the domain border tends to maintain circular symmetry, surface-viscous stresses generated by surface compression tend to destabilize the perimeter. A shape instability arises above a critical surface capillary number (Ca) relating surface-viscous stresses to line tension. Moreover, we show that the mechanism of instability is distinct from that of the famous Saffman-Taylor fingering instability. Various extensions of our model are discussed, including materials with finite dilatational surface viscosity, linear and nonlinear (visco)elasticity, and large-amplitude deformations.

Main Flow Migration in the Middle Yangtze River Influenced by Cascade Reservoirs: Characteristics, Controlling Factors, Trends, and Ecological Impact

The main flow migration in the middle Yangtze River occurs in most river sections and is affected by factors such as incoming water and sediment, riverbed boundaries, and channel shapes, leading to a complex riverbed evolution. Revealing the controlling factors and analyzing the developmental trends are important for addressing the adverse ecological impacts caused by these changes. Based on a large amount of observational data since the impoundment of the Three Gorges Reservoir, the characteristics of the main flow migration in the middle Yangtze River under different flow conditions were analyzed, and its correlation with the nodes and bars at the inlet, the plane shape of the river, and riverbed morphology were determined to identify the key controlling factors. The results showed that it is characterized by the displacement of the main flow zone during the middle-flow period. The key factors controlling the main flow migration include the deflecting action of the nodes and sidebars at the inlet, relaxation of the channel plane shape, and resistance difference caused by the riverbed morphology between the branches. The trend analysis suggests that the main flow migration in the middle Yangtze River may become more frequent after the operation of the cascade reservoirs in the future and may threaten the ecological environment.

Understanding the Rheology of Polymer–Polymer Interfaces Covered with Janus Nanoparticles: Polymer Blends versus Particle Sandwiched Multilayers

Interfacial rheology is crucial in dictating morphology and ultimate properties of particle-stabilized polymer blends, but is challenging to be determined. In this study, a fully polymeric dumbbell-shaped Janus nanoparticle (JNP) of polymethyl methacrylate (PMMA) and polystyrene (PS) spheres with equal sizes (∼80 nm) was prepared and used as an efficient compatibilizer for PMMA/PS blends. The JNPs were preferentially localized at the PMMA/PS interface, thereby reducing the interfacial tension and refining the morphology in both droplet-matrix and co-continuous type blends, whereby a JNP concentration ∼2.5 wt % is sufficient to reach a saturation in droplet size reduction due to compatibilization. Based on the linear viscoelastic moduli and corresponding relaxation spectra (H(τ)*τ) of JNP-compatibilized droplet-matrix blends, besides the droplet shape relaxation time (τF), a longer relaxation time (τβ), typically related to interfacial viscoelasticity, was readily identified. The dependence of τβ on the JNP concentration (WJNPs) was significantly dominated by the droplet size reduction induced by the JNP compatibilization, with τβ decreasing with increasing WJNPs. The viscoelastic properties extracted from τβ typically originate from a combination of gradients in interfacial tension due to the particle redistribution at the droplet interface (Marangoni stresses) and the deviatoric stresses of intrinsic rheological origin. The latter originate from the intrinsic viscoelasticity of the particle-laden interface, which is enhanced by particle jamming and particle–polymer interactions, such as entanglements between chains from the polymeric spheres and those penetrating from the bulk into the spheres. To address the challenge of isolating these contributions, a JNP-sandwiched PMMA/PS multilayer structure was designed to exclude the effect of Marangoni stresses and droplet curvature, thus having no τF but a new relaxation (τ′β), which characterizes the contribution of intrinsic interfacial viscoelasticity. The τ′β was observed to increase with JNP coverage (Σ) following the Vogel–Fulcher–Tammann model that is typically used to describe the divergent behavior of the “cage” effect in classical colloidal glasses. Moreover, a multimode Maxwell model fitting allows to split the interfacial relaxation into the confined diffusion of JNPs within their cage and the entanglements between the JNPs and the bulk.

Surface stress and shape relaxation of gelling droplets.

Solidification is a heterogeneous transformation from liquid to solid, which usually combines transport, phase transition and mechanical strain. Predicting the shapes resulting from such a complex process is fascinating and has a wide range of implications from morphogenesis in biological tissues to industrial processes. For soft solids initially at equilibrium, elastic stresses, whether tensile or compressive, can be induced by heterogeneous volumetric deformations of the material. These stresses trigger surface instabilities leading to variations of curvature and shape of the solids. In this article, we study the shape evolution of elongated droplets of polymer and particle suspensions undergoing a solidification process caused by the inward diffusion of a gelling agent from the surface. We show experimentally and numerically that there appears a layer of gelled material growing at the surface. Due to volume contraction, this layer induces tensile stresses and drives a flow in the ungelled liquid core, resulting in the relaxation of the droplets toward spherical shapes. Over time, the thickness of this elastic membrane grows, hence the bending stiffness required to change its shape eventually balances the surface stresses, which arrests the relaxation process. These results provide general rules to understand the shape of solidifying materials combining both tension and bending driven deformations.

Coupling cell shape and velocity leads to oscillation and circling in keratocyte galvanotaxis

During wound healing, fish keratocyte cells undergo galvanotaxis where they follow a wound-induced electric field. In addition to their stereotypical persistent motion, keratocytes can develop circular motion without a field or oscillate while crawling in the field direction. We developed a coarse-grained phenomenological model that captures these keratocyte behaviors. We fit this model to experimental data on keratocyte response to an electric field being turned on. A critical element of our model is a tendency for cells to turn toward their long axis, arising from a coupling between cell shape and velocity, which gives rise to oscillatory and circular motion. Galvanotaxis is influenced not only by the field-dependent responses, but also cell speed and cell shape relaxation rate. When the cell reacts to an electric field being turned on, our model predicts that stiff, slow cells react slowly but follow the signal reliably. Cells that polarize and align to the field at a faster rate react more quickly and follow the signal more reliably. When cells are exposed to a field that switches direction rapidly, cells follow the average of field directions, while if the field is switched more slowly, cells follow a “staircase” pattern. Our study indicated that a simple phenomenological model coupling cell speed and shape is sufficient to reproduce a broad variety of different keratocyte behaviors, ranging from circling to oscillation to galvanotactic response, by only varying a few parameters.

Shape and structural relaxation of colloidal tactoids

Facile geometric-structural response of liquid crystalline colloids to external fields enables many technological advances. However, the relaxation mechanisms for liquid crystalline colloids under mobile boundaries remain still unexplored. Here, by combining experiments, numerical simulations and theory, we describe the shape and structural relaxation of colloidal liquid crystalline micro-droplets, called tactoids, where amyloid fibrils and cellulose nanocrystals are used as model systems. We show that tactoids shape relaxation bears a universal single exponential decay signature and derive an analytic expression to predict this out of equilibrium process, which is governed by liquid crystalline anisotropic and isotropic contributions. The tactoids structural relaxation shows fundamentally different paths, with first- and second-order exponential decays, depending on the existence of splay/bend/twist orientation structures in the ground state. Our findings offer a comprehensive understanding on dynamic confinement effects in liquid crystalline colloidal systems and may set unexplored directions in the development of novel responsive materials.

Adjustable brittle-ductile transition behavior and rheological behavior of polypropylene random copolymer nanocomposites through well interfacial-loaded nanoparticles

To improve the low-temperature toughness of polypropylene random copolymer (PPR), a series of PPR/ethylene-propylene rubber/silica (PPR/EPR/SiO2) nanocomposites were prepared through toughening strategy based on interfacial-loaded particles and adjustable dispersed phase. The results showed that the brittle-ductile transition behavior for PPR/EPR/SiO2 nanocomposites could be adjusted by silica and rubber contents, and consequently the composites were endowed with excellent low-temperature impact resistance. Furthermore, the nanocomposites never presented the significant deterioration of rigidity compared with pure PPR. The dynamic rheological results showed that viscosity change in PPR matrix and EPR phase could make silica particles preferentially locate at PPR-EPR interface, which strongly reduced the size of EPR domain and consequently led to a stronger shape relaxation of long-time scale. According to dynamic rheological behavior, microstructure and interparticle distance analysis of PPR nanocomposites, the adjustable brittle-ductile transition behavior was ascribed to the reduced rubber interparticle distance coming from the well interfacial-loaded nanoparticles.

High-Temperature Quantum Tunneling and Hydrogen Bonding Rearrangements Characterize the Solid-Solid Phase Transitions in a Phosphonium-Based Protic Ionic Liquid.

We report the complex phase behavior of the glass forming protic ionic liquid (PIL) d3‐octylphosphonium bis(trifluoromethylsulfonyl)imide [C8H17PD3][NTf2] by means of solid‐state NMR spectroscopy. Combined line shape and spin relaxation studies of the deuterons in the PD3 group of the octylphosphonium cation allow to map and correlate the phase behavior for a broad temperature range from 71 K to 343 K. In the solid PIL at 71 K, we observed a static state, characterized by the first deuteron quadrupole coupling constant reported for PD3 deuterons. A transition enthalpy of about 12 kJ mol−1 from the static to the mobile state with increasing temperature suggests the breaking of a weak, charge‐enhanced hydrogen bond between cation and anion. The highly mobile phase above 100 K exhibits an almost disappearing activation barrier, strongly indicating quantum tunneling. Thus, we provide first evidence of tunneling driven mobility of the hydrogen bonded P−D moieties in the glassy state of PILs, already at surprisingly high temperatures up to 200 K. Above 250 K, the mobile phase turns from anisotropic to isotropic motion, and indicates strong internal rotation of the PD3 group. The analyzed line shapes and spin relaxation times allow us to link the structural and dynamical behavior at molecular level with the phase behavior beyond the DSC traces.

Purified myelin lipids display a critical mixing point at low surface pressure

Lipids extracted from Purified Myelin Membranes (LPMM) were spread as monomolecular films at the air/aqueous interface. The films were visualized by Brewster Angle Microscopy (BAM) at different lateral pressures (π) and ionic environments. Coexistence of Liquid-Expanded (LE) and cholesterol-enriched (CE) rounded domains persisted up to π ≈ 5 mN/m but the monolayers became homogeneous at higher surface pressures. Before mixing, the domains distorted to non-rounded domains. We experimentally measured the line tension (λ) for the lipid monolayers at the domain borders by a shape relaxation technique using non-homogeneous electric fields. Regardless of the subphase conditions, the obtained line tensions are of the order of pN and tended to decrease as lateral pressure increased toward the mixing point. From the mean square displacement of nested trapped domains, we also calculated the dipole density difference between phases (μ). A non-linear drop was detected in this parameter as the mixing point is approached. Here we quantitively evaluated the π-dependance of both parameters with proper power laws in the vicinity of the critical mixing surface pressure, and the exponents showed to be consistent with a critical phenomenon in the two-dimensional Ising universality class. This idea of bidimensionality was found to be compatible only for simplified lipidic systems, while for whole myelin monolayers, that means including proteins, no critical mixing point was detected.Finally, the line tension values were related with the thickness differences between phases (Δt) near the critical point.

Application of X-ray Microcomputed Tomography for the Static and Dynamic Characterization of the Microstructure of Oleofoams.

Oleofoams are a novel, versatile, and biocompatible soft material that finds application in drug, cosmetic or nutraceuticals delivery. However, due to their temperature-sensitive and opaque nature, the characterization of oleofoams’ microstructure is challenging. Here, synchrotron X-ray microcomputed tomography and radiography are applied to study the microstructure of a triglyceride-based oleofoam. These techniques enable non-destructive, quantitative, 3D measurements of native samples to determine the thermodynamic and kinetic behavior of oleofoams at different stages of their life cycle. During processing, a constant bubble size distribution is reached after few minutes of shearing, while the number of bubbles incorporated keeps increasing until saturation of the continuous phase. Low amounts of solid triglycerides in oleofoams allow faster aeration and a more homogeneous microstructure but lower thermodynamic stability, with bubble disproportionation and shape relaxation over time. Radiography shows that heating causes Ostwald ripening and coalescence of bubbles, with an increase of their diameter and sphericity.

Uncovering associations between data-driven learned qMRI biomarkers and chronic pain

Knee pain is the most common and debilitating symptom of knee osteoarthritis (OA). While there is a perceived association between OA imaging biomarkers and pain, there are weak or conflicting findings for this relationship. This study uses Deep Learning (DL) models to elucidate associations between bone shape, cartilage thickness and T2 relaxation times extracted from Magnetic Resonance Images (MRI) and chronic knee pain. Class Activation Maps (Grad-CAM) applied on the trained chronic pain DL models are used to evaluate the locations of features associated with presence and absence of pain. For the cartilage thickness biomarker, the presence of features sensitive for pain presence were generally located in the medial side, while the features specific for pain absence were generally located in the anterior lateral side. This suggests that the association of cartilage thickness and pain varies, requiring a more personalized averaging strategy. We propose a novel DL-guided definition for cartilage thickness spatial averaging based on Grad-CAM weights. We showed a significant improvement modeling chronic knee pain with the inclusion of the novel biomarker definition: likelihood ratio test p-values of 7.01 × 10–33 and 1.93 × 10–14 for DL-guided cartilage thickness averaging for the femur and tibia, respectively, compared to the cartilage thickness compartment averaging.

Quantitative prediction of elongation deformation and shape relaxation of a red blood cell under tensile and shear stresses

The characteristic time of stress relaxation is a key viscoelastic property of cell membrane that controls time-dependent processes such as shape recovery. Although many experimental studies have been devoted to the measurement of characteristic relaxation time, considerable uncertainty still stands because existing methods rely on different experimental designs and analyses. Here, we present a mesoscopic computational study to investigate the elastic deformation and relaxation characteristics of an isolated red blood cell (RBC) under both tensile and shear stresses. We examine the elastic response and relaxation behavior of the RBC under static tensile stretching and dynamic shear stress. When the cell deformation index responding fluid shear stress is equivalent to the one responding external tensile stretching, we find that the characteristic relaxation time for the RBC in planar flows is longer than that for the RBC under tensile stretching. We also subject the RBC to confined tube/channel flows to probe the effect of geometric confinement on its elastic deformation and relaxation dynamics. Our simulations show that the computed characteristic relaxation time is further increased when compared to those obtained under tensile stretching or planar flows, indicating that the confinement would slow down the cell relaxation process, especially under strong confinement conditions. These findings may facilitate a better understanding of variable relaxation time observed in different experiments.

Ionic environment, thickness and line tension as determinants of phase separation in whole Purified Myelin Membranes monolayers

Purified myelin membranes (PMM) were spread as monomolecular films at the air/aqueous solution interface, and visualized by Brewster Angle Microscopy (BAM) at different lateral pressures (π) on three specific aqueous solutions: absence of salts, physiological conditions and presence of calcium. Coexistence of Liquid-Expanded (LE) and Liquid Ordered (LO) phases persisted up to collapse in the presence of salts, whereas monolayers became homogeneous at π ≥ 35−40 mN/m when salts are absent. This PMM phase-mixing behavior in monolayers is similar to the previously reported behavior of PMM multilamellar vesicles. Reflectivities (Rp) of p-polarized light from both phases were assessed throughout the whole π -range, and film thicknesses (t) were calculated from the Rp values and measured film refractive indices (n). The LO phase was found to be more reflective and thicker than the LE phase at π ≤ 15 mN/m, but less reflective and thinner at higher π.We also determined the line tension (λ) of PMM monolayers at the domain boundaries from the rate of domain shape relaxation, which turned out to be of the order of picoNewtons (pN) and decreased as π increased. A correlation between λ and thickness differences (Δt) was found, suggesting that Δt is a molecular determinant for λ in PMM monolayers.Both λ and Δt were found to increase markedly when calcium was present in the subphase. This result corroborates the concept of divalent cations as a stabilizing factor for phase separation, in line with earlier studies on this mixture forming multilamellar membrane arrangements.

Isostasy with Love: II Airy compensation arising from viscoelastic relaxation

SUMMARY In modern geodynamics, isostasy can be viewed either as the static equilibrium of the crust that minimizes deviatoric stresses, or as a dynamic process resulting from the viscous relaxation of the non-hydrostatic crustal shape. Paper I gave a general formulation of Airy isostasy as an elastic loading problem solved with Love numbers, and applied it to the case of minimum stress isostasy. In this sequel, the same framework is used to study Airy isostasy as the long-time evolution of a viscoelastic shell submitted to surface and internal loads. Isostatic ratios are defined in terms of time-dependent deviatoric Love numbers. Dynamic isostasy depends on the loading history, two examples of which are the constant load applied on the surface in the far past and the constant shape maintained by addition or removal of material at the compensation depth. The former model results in a shape decreasing exponentially with time and has no elastic analogue, whereas the latter (stationary) model is equivalent to a form of elastic isostasy. Viscoelastic and viscous approaches are completely equivalent. If both load and shape vary slowly with time, isostatic ratios look like those of the stationary model. Isostatic models thus belong to two independent groups: the elastic/stationary approaches and the time-dependent approaches. If the shell is homogeneous, all models predict a similar compensation of large-scale gravity perturbations. If the shell rheology depends on depth, stationary models predict more compensation at long wavelengths, whereas time-dependent models result in negligible compensation. Mathematica and Fortran codes are available for computing the isostatic ratios of an incompressible body with three homogeneous layers.

NMR investigation on the honeycomb iridate Ag3LiIr2O6

We investigate the structural and magnetic properties of a Kitaev spin liquid candidate material Ag$_3$LiIr$_2$O$_6$ based on $^7$Li nuclear magnetic resonance line shape, Knight shift and spin-lattice relaxation rate $1/T_1$. The first sample A shows signatures of magnetically ordered spins, and exhibits one sharp $^7$Li peak with FWHM increasing significantly below 14~K. $1/T_1^{stretch}$ of this sample displays a broad local maximum at 40~K, followed by a very sharp peak at $T_N = 9\pm1$~K due to critical slowing down of Ir spin fluctuations, a typical signature of magnetic long range order. In order to shed light on the position-by-position variation of $1/T_1$ throughout the sample, we use Inverse Laplace Transform $T_1$ analysis based on Tikhonov regularization to deduce the density distribution function $P(1/T_1)$. We demonstrate that $\sim 60\%$ of Ir spins are statically ordered at the NMR measurement timescale but the rest of the sample volume remains paramagnetic even at 4.2~K, presumably because of structural disorder induced primarily by stacking faults. In order to further investigate the influence of structural disorder, we compare these NMR results with those of a second sample B, which has been shown by transmission electron microscope to have domains with unwanted Ag inclusion at Li and Ir sites within the Ir honeycomb planes. The sample B displays an additional NMR peak with relative intensity of $\sim 17\%$. The small Knight shift and $1/T_1$ of these defect-induced $^7$Li sites and the enhancement of bulk susceptibility at low temperatures suggest that these defects generate domains of only weakly magnetic Ir spins accompanied by free spins, leading to a lack of clear signatures of long-range order. The apparent lack of long-range order could be easily misinterpreted as evidence for the realization of a spin liquid ground state in highly disordered Kitaev lattice.