Papers

The accurate estimation of the ionic activity coefficients is paramount for the characterization of the electrochemical reactions driving energy storage processes as well as the biological mechanisms involving ATP hydrolysis. The numerical evaluation of the ionic activity coefficients by computational methods is rendered difficult by the infinite dilution limit involved in the integral identity between the Helmholtz free energy and the equation of state of the electrolyte. By combining a recently introduced self-consistent Debye-Huckel formalism and its analytically tractable low density limit, we develop a computationally efficient approach enabling the accurate calculation of the free energy and the related thermodynamic functions of aqueous electrolytes over a broad range of ionic concentration and electrostatic coupling strength. Via comparison with numerical simulations and experimental data, it is shown that our approach can predict the salt dependence of the Helmholtz free energy, the entropy, and the ionic activity coefficients from the electrostatically driven submolar ion concentrations to the hard-core dominated molar density regime.

Fracture in polymer networks is strongly influenced by how network architecture and topology organize damage in space. Although these materials can sustain large deformations under homogeneous loading, they often fail in a brittle and flaw-sensitive manner once defects are present. Here we show that network heterogeneity is one key architectural parameter controlling this transition. Using a discrete mesoscale network model that explicitly resolves strand-level prestretch, force transmission, and irreversible bond scission, we demonstrate that heterogeneity in prestretch and strand lengths broadens crack-induced deformation, delocalizes bond rupture, and enlarges the fracture process zone. In polydisperse networks, increasing contour-length disorder broadens the prestretch distribution and systematically increases both toughness and the intrinsic length scale governing flaw sensitivity. Bimodal networks amplify this mechanism further through combined contrasts in prestretch and strand lengths, producing the largest damage zones and the highest toughness among the architectures studied. Inspired by the results, we derive a scaling estimate relating the emergent fracture-active length to the spatial range of crack-induced deformation and the logarithmic spread of strands' remaining stretch to rupture.

We investigate a minimal visual perception-driven model of self-propelled particles that interact through purely position-based cohesive steering within a finite vision cone, in the absence of velocity alignment and physical volume exclusion. Building on a previously proposed minimal cognitive model, we extend the interaction range from the previously studied short-range and distance-independent limits to an intermediate regime, enabling a systematic investigation of how a finite vision range influences collective behavior. We find that this regime exhibits a rich state diagram that combines characteristics of both limiting cases while giving rise to qualitatively new collective states. Besides the gas, nematic-band, and polar file states, we uncover the emergence of active foams – system-spanning fractal-like networks sustained by dynamically evolving particle streams. These structures arise solely from non-reciprocal vision-based interactions and are absent in both the short and long vision-range limits. For larger vision angles, the system self-organizes into a variety of aggregate morphologies, including condensates, rings, and multi-twist filaments, revealing a strong interplay between interaction range and anisotropic perception in shaping collective organization.

A covalent polymer network is orders of magnitude weaker than individual covalent bonds. This large gap reveals an unrealized potential. The weakness of a network comes not from individual covalent bonds, but from how they are connected. Here we unlock this potential by identifying a topological criterion, involving optimally sparse crosslinks, long chains, and dense entanglements. Optimizing each topological variable alone is insufficient for high strength, yet exceptional strength emerges when they are combined within a narrow regime. We describe the physics of this synergy, and use the criterion to strengthen networks widely used but known to be weak. For example, we prepare a polyacrylamide hydrogel of 90 wt% water, amplifying strength from a commonly reported value of ~0.1 MPa to ~6.5 MPa. The exceptional strength is achieved without sacrificing other properties, such as softness, toughness, and elasticity. The criterion establishes topology as a general design principle for strong polymer networks.

We formulate the Generalized Constraint-Controlled Rheological Model (GCCRM) for entangled linear and long-arm-branched polymer melts under sinusoidal periodic shear deformation (SPSD), spanning the linear and strongly nonlinear regimes. Molecular architecture, conformation, chain stretch, flow-active constraints, convective constraint release, dynamic tube dilation, and activated arm retraction define a common upstream molecular state. This state feeds two time-domain constitutive pathways (a configurational stress and a dissipative internal-variable stress) which are recombined before Fourier analysis. For branched polymers, the architecture is described by an entire backbone of molecular weight $M_{bb}$ and $N_a$ long arms of molecular weight $M_a$, with a compliance-weighted representative-constitutive-cell partition. The linear limit is normalized so that the configurational moduli sum to the entanglement modulus, while dynamic tube dilation and stretch drive distinct contributions to dissipative activation. Under reduced closures, GCCRM generates a smooth nonlinear elastic response with an onset of nonlinearity, an approximately logarithmic growth regime, an architecture-dependent maximum, and distinct high-deformation asymptotes. Fourier harmonics, ELR, VLR, and Lissajous curves are treated as post-processing observables of the total stress. The framework establishes a consistent constitutive architecture whose quantitative validation requires one parameter set per material across SPSD and, prospectively, transient shear histories.

Ionic liquids are materials whose properties can be tailored for a wide range of applications, but their chemical diversity and complex electronic structure make systematic screening using quantum-mechanical methods computationally demanding. Semiempirical quantum methods offer an alternative, although their accuracy depends on their parameterization. We investigate the reparameterization of the DFTB method GFN1-xTB for predicting structural, energetic, and electronic properties of ionic liquids. For this purpose, we built a database containing over 14,000 conformations to construct training datasets and explored different parameterization strategies fitted to binding energies, forces, HOMO-LUMO gaps, and geometries. Analysis of the original GFN1-xTB model reveals systematic errors in these properties, with distinct behavior for saturated and aromatic cations. Thus, we investigate two models: CXMIM-xTB, a targeted model trained on imidazolium-based ionic liquids, and IL-xTB, a more general model designed to describe systems containing saturated and aromatic cations. CXMIM-xTB improves the description of binding energies, geometries, and frontier orbital energies while retaining computational efficiency. The model is subsequently applied to estimate electrochemical windows of ionic liquids, providing values consistent with DFT and experimental values. IL-xTB follows a hierarchical parameterization strategy including different cation and anion families and substantially reduces the systematic errors of GFN1-xTB across the chemical space considered, providing a route for developing xTB parameterizations with broader applicability to ionic liquids. Overall, both parameterization strategies outperform the original GFN1-xTB model, demonstrating that reparameterized methods provide an efficient approach for connecting configurational sampling with quantum-mechanical properties of ionic liquids.

Slender structures in dense environments interact through contact, which alters how and when they become unstable. Here we show that contact makes snap-through history dependent, so that when and whether a beam pair snaps depends on how long the contact is loaded. Under cyclic compression, the pair snaps on loading, snaps on unloading, or stays packed, with the loading and unloading rates selecting among the three. Upon contact, one beam rolls along the other without slip, and faster driving lets it roll further before it slides and snaps: to larger compression on loading and to smaller compression on unloading. This delay is asymmetric between loading and unloading as the unloading branch carries a memory of the loading rate. Interestingly, above a threshold compression the pair snaps even when the compression is stopped, with the time to snap diverging at that threshold. By exploiting this asymmetric, rate-dependent delay, we program ordered sequences of snap events in arrays of pairs with tunable gaps, and demonstrate that the energy dissipated per cycle becomes a two-parameter function of the loading and unloading rates, increasing fourfold when, at the fastest loading, the return is slowed.

The linear viscoelastic behavior of a polymer shows an equivalence of loading frequency and temperature, the time-temperature equivalence (TTE). Horizontal shifting extends moduli measured over limited ranges to much wider ranges, through the glass transition. Multimaterial 3D printing has created digitally mixed polymers, a new class in which elastomeric and glassy resins are mixed voxel by voxel in prescribed ratios. Composition shifts the location and breadth of the glass transition and increases the rubbery modulus. Here we extend TTE to a time-temperature-composition equivalence (TTCE). Across all measured temperatures, frequencies, and compositions, the storage modulus collapses onto one master curve. Composition translates the transition and stretches its breadth, both linearly. The phase angle and relaxation spectrum superpose under the same operations, so the family shares one distribution of relaxation times. A single closed-form TTCE gives the storage and loss moduli of the family, guiding graded and architected designs for elasticity and dissipation.

Soft matter systems often display shape transitions, mode competition, and threshold-like dynamical responses driven by the coupling between mechanics, geometry, and biochemical or active fields. Here we present a minimal theoretical framework for describing such morphodynamic selection using a small number of physically interpretable order parameters. The model combines elastic relaxation, active driving, and nonlinear feedback, yielding distinct regimes corresponding to weak deformation, single-mode selection, and coupled shape transformation. We derive the governing equations, identify the relevant dimensionless control parameters, and show how the phase behavior can be organized in terms of threshold competition and effective coupling strengths. The framework provides a compact route for connecting analytical theory, numerical simulations, and experimentally observable shape dynamics in soft and biological matter.

Linear stability analysis predicts the onset of wrinkling, while weakly nonlinear analysis determines whether the bifurcation is supercritical or subcritical through the sign of the cubic amplitude coefficient. When this coefficient vanishes, higher-order terms are needed. In this paper, we develop a higher-order Stroh framework for weakly nonlinear post-bifurcation analysis of finitely deformed elastic solids. For each nonzero harmonic, the same base-state Stroh operator is retained at every order, with nonlinear source terms assembled from lower-order fields to account for geometry, constitutive laws and constraints. This recursive structure, together with Fredholm solvability conditions, enables us to derive cubic, quintic and higher-order amplitude equations. Here, we demonstrate the power of this framework by analysing wrinkling in an incompressible neo-Hookean film bonded to an exponentially graded substrate. The results recover established linear instability benchmarks and the supercritical-subcritical transition in the homogeneous-substrate limit. For graded substrates, the cubic coefficient vanishes on a surface in the parameter space of modulus ratio and substrate gradation. On this surface, the quintic coefficient takes both signs, corresponding to supercritical and subcritical bifurcations. These regions are separated by a curve along which the quintic coefficient also vanishes. Determining the branch direction along this curve requires higher-order terms, which can be calculated within the same framework.

Natural habitats confront microorganisms with heterogeneous, mechanically complex surroundings that shape how they move and explore. Most studies of motility in complex media treat the environment as either a homogeneous fluid or a rigid obstacle field. Here we instead study a medium that is simultaneously soft and reconfigurable — deformable by the organism moving through it. We track the unicellular biflagellate microalga Chlamydomonas reinhardtii swimming in a soft, disordered suspension of microgel particles comparable in size to the cell itself, across a range of gel packing fractions. With increasing packing fraction, trajectories change from long straight runs to short, frequently reorienting ones, and the effective translational diffusivity decreases monotonically. The bimodal run/tumble speed distribution, characteristic of the free-swimming gait, collapses toward a single mode as the packing fraction increases, by near-proportional slowing of both gait states rather than a redistribution of cells between them. Recasting trajectories as alternating hops and traps reveals the signature of a deformable medium: hop lengths halve as the gel densifies, approaching the microgel aggregate scale, while trapping times do not increase with increasing packing fraction. Independent force estimates place the cell's propulsive force close to an order of magnitude above that needed to deform a single grain, so remodeling is expected rather than merely observed. A minimal active remodeler model — an active Brownian particle among mobile, rigid obstacles — shows that obstacle mobility alone is sufficient to relocate the onset of permanent trapping. Thus, our results establish obstacle mobility as a control parameter for motility in soft, disordered matter.

Granular surface flows are common in industrial practice and natural systems, however, theoretical description of such flows is at present incomplete. Two prototype systems involving surface flow are compared: heap formation by pouring at a point and rotating cylinders. Continuum models for analysis of these flows are reviewed, and experimental results for quasi-2d systems are presented. Experimental results in both systems are well described by continuum models.

Self-propelled Janus particles are promising active agents for microscale manipulation and microrobotic applications. However, when powered by a global external energy source, such as an electric, magnetic, or optical field, all particles respond simultaneously, preventing their independent and concurrent control. Achieving parallel, decoupled control without continuous particle-specific feedback, while preserving intrinsic self-propulsion, remains a significant challenge. Here, we present an optoelectronic (OE) strategy for simultaneously and independently controlling electrically driven, self-propelled metallo-dielectric Janus particles (JPs) using spatially programmable virtual electrodes. These electrodes are generated by projecting a single predefined illumination pattern from a digital micromirror device onto a photoconductive surface. The resulting confinement landscape enables multiple JPs to follow distinct trajectories without continuous monitoring, feedback, or pattern reconfiguration. We demonstrate controlled pairing of JPs with passive polystyrene particles and biological cells. For automated multi-agent operation, an image-based assignment framework identifies optimal agent-target pairs, excludes intersecting trajectories, and generates a single composite pattern for simultaneous, selective pair formation without real-time updating. Finally, we demonstrate independent manipulation of multiple magnetically rolling JPs under a globally applied magnetic field. Overall, this programmable open-loop strategy provides a versatile framework for multi-agent microrobotic manipulation, with potential applications in in-vitro micromanipulation and microfluidic systems.

Searches for dense packings of congruent shapes move rigid pieces. Hardening starts each piece as its inscribed disk and deforms it into the target shape while the container is compressed. We ask whether it is a search worth running alongside existing ones, and test it at matched budget and comparable tuning against rigid starts, growth, simulated annealing and perturbation–compression on seven families of squares, triangles, hexagons and disks in squares, triangles and circles (188 instances), scoring each method by the lowest container size it reaches. Hardening is not a better default: tuned rigid starts reach the lowest size most often. It is a different search. On squares in a square it reaches packings the other four methods miss, including Trump's n = 11 and Wainwright's n = 19 in 32 runs per method, and in a targeted follow-up campaign at ten times the budget (16 runs per method) it is the only one to reach the best known packings for n = 17 and n = 26, while annealing alone reaches n = 27. In exploratory ablations, compressing disks and then switching to polygons at once, or growing rigid polygons along hardening's area schedule, shows no detectable difference from rigid starts, and hardening's unique packings contain more tilted pieces. The bias costs elsewhere: hardening is worse on triangles and on four families held out from all development, where the short-pilot policy we tested showed no detected advantage over always starting rigid. Shape continuation is an alternative search that reaches different packings, worth keeping beside rigid ones; we did not show that splitting a fixed budget between them pays, and we have no test that predicts where it helps. Search campaigns were planned before they ran, with amendments, exploratory analyses and post-hoc checks identified separately; every run is replayable and every number is generated from an event log.

Biomolecular condensates are micron-scale, membraneless compartments that concentrate proteins and nucleic acids by liquid-liquid phase separation (LLPS). Condensates play critical roles in transcription, mRNA metabolism, stress responses, and signaling across the cell, with their dysregulation increasingly correlated with several pathologies spanning neurodegeneration, cancer, cardiometabolic disease, and viral infections. The emergent biology of condensates can be cohesively viewed through the framework of colloidal, interfacial, and transport phenomena. Here we review the condensate literature through this joint lens, treating each condensate as a biological colloid whose behavior is captured by four measurable physical quantities: a Flory–Huggins interaction parameter that determines whether the dense phase forms at all, a partition coefficient that dictates the degree of client molecule accumulation, a viscoelastic relaxation time that sets whether the dense phase responds as a liquid or a solid, and an interfacial tension that governs droplet shape and interfacial dynamics. Each quantity is traced to its physical origin, its measured range in biological condensates, and the techniques that quantify it, for both three-dimensional droplets and two-dimensional membrane-tethered condensates. Interpreting disease as the displacement of these physical quantities from their functional ranges, we reorganize condensate pathologies into a small set of physical failure modes — saturation concentration dysregulation, aberrant aging and liquid-to-solid transitions, disrupted partitioning, interfacial aggregation, and pathogen hijacking — and show how the same parameters become design variables for condensate-targeted drugs and engineered coacervate delivery vehicles. The review concludes with open questions at the interface of condensate biology and colloidal and interfacial science.

In most theories of electrolyte and polyelectrolyte solutions, the solvent is treated implicitly as a dielectric continuum. While this approximation has enabled major advances, it neglects coupling between ionic and dipolar degrees of freedom. Using molecular dynamics simulations and the mean-spherical approximation, we show that this coupling qualitatively alters electrostatically driven phase behaviour. At finite ion concentrations, many-body ion-dipole correlations reverse the temperature dependence of effective electrostatic interactions, leading to upper-critical solution temperature (UCST) behaviour in direct contrast to the lower-critical solution temperature (LCST) behaviour predicted by implicit-solvent electrostatic theories. This phase reversal arises from ion-induced suppression of local dipolar screening, a collective effect inaccessible to pairwise implicit-solvent descriptions.

Self-assembled monolayers (SAMs), formed by the adsorption of a monomolecular layer of adsorbate onto a surface, have emerged as leading candidates for plat- forms for electrochemical biosensors. In order to characterize the working age of a SAM-based biosensor, it is necessary to understand the long-term degradation of the monolayer, with particular emphasis on the intermolecular forces present in the self-assembly process. In this work, discrete- and continuous-time proba- bilistic cellular automaton (PCA) models for the degradation of SAMs involving adsorbate-adsorbate interactions are proposed in which adsorbate molecules desorb at a rate dependent only on the number of adsorbed neighbors. Impor- tantly, we make no assumptions on the relationship between degradation rate and adsorbed neighbors. Using techniques from generating function theory and complex analysis, the expected degradation curve is solved for explicitly in one dimension and shown to be a sigmoid curve related to the Gompertz function.

Magnetic actuation scales with a particle's magnetic moment, yet practical colloidal suspensions remain superparamagnetic or weakly magnetized, a permanent dipole drives a near contact attraction growing as the square of the moment, binding particles irreversibly. Here we show that this constraint depends not only on how much magnetic material a particle carries, but on where it sits. Partitioning a fixed moment M into N internal elements leaves the far field interaction, and every actuation force, unchanged while suppressing the near-field pair energy by 1/N. As we show in our experiments across 57 samples and three geometries, stability collapses onto a single $\sim M^{2}/N$ boundary, raising the accessible dipolar energy scale nearly 150-fold. A liquid-phase analogue of magnetic force microscopy images the engineered energy landscape directly. The resulting colloids outrun size-matched superparamagnetic beads in weak fields far from a magnetic source, suggesting their applicability in hard to reach places. Together, these results open a new operating regime for colloidal manipulation, with impactful implications in microfluidics and drug delivery.

We investigate the rate of change of equilibrium potential energy for nonlinear elastic solids undergoing evolutions that may involve changes in body geometry, material properties, applied tractions, and prescribed displacements. We represent such an evolution by a one-parameter family of variational problems and identify a class of these families for which the potential-energy release rate is well defined. Particular attention is given to the formulation of the potential energy at each evolution state: each constituent variational problem must yield the appropriate equilibrium state, while its equilibrium value must also correctly represent the physical change in potential energy between states. Within this framework, we derive a closed-form expression for the potential-energy release rate for solids with bounded domains, allowing for spatially nonuniform material evolution, sharp material interfaces, and evolving boundary geometry and prescribed boundary data. The expression is determined entirely by the equilibrium solution at the state under consideration and by the quantities prescribing the local evolution, so neighboring equilibrium problems need not be solved. Finally, we apply the result to the extension of a notch in a two-dimensional elastic body and show that the resulting potential-energy release rate reduces to the classical $J$-integral.

Frustration generated by competing interactions can stabilise patterns with tunable lengthscales which evade indefinite coarsening and macroscopic phase separation. Here we investigate a charged binary fluid in which the tendency toward phase separation is opposed by screened electrostatic repulsion. Combining analytics and numerical simulations, we identify a regime of arrested phase separation characterised by finite-wavelength instability. The resulting states include droplet and labyrinthine phases in two dimensions, and droplets, concentric shells and tubular structures in three dimensions. Surprisingly, some of these structures are amorphous morphologies that emerge reproducibly from different initial conditions. Free-energy estimates suggest that these disordered structures may be thermodynamically stabilised by frustration, rather than being kinetically trapped. Our results identify a simple mechanism through which the competition between phase separation and screened electrostatics can generate equilibrium amorphous microphases with glassy features.

A plate withdrawn from a bath of partially wetting liquid entrains a film above a critical velocity. Here, we show that this critical threshold can be substantially increased by the presence of a laminar gas flow opposed to the plate velocity that exerts a tangential shear stress on the liquid-gas interface. Using two-phase immiscible Navier-Stokes simulations and confining the gas to a narrow channel, we derive a closed-form expression for the interfacial stress, which combined with lubrication theory relates the thickness of the film and the contact line velocity to the applied shear stress. As the gas velocity is increased, the film thins and the contact line velocity slows down, until the imposed shear overcomes the drag of the moving plate and entrainment is arrested. As in the quiescent case, the velocity mismatch between plate and contact line is independent of the plate velocity and is set by the applied shear alone. We find that one set of varying gas flow simulations, with constant critical plate velocity, therefore determines the critical gas flow for every smaller plate velocity.

Transport networks are key to living and engineered systems. How can transport networks adapt efficiently using only local rules, without a global objective or specific boundary conditions? Inspired by observations in biological transport networks, we adapt networks by subsequently pruning edges by relative resistance, thus removing edges whose resistance is higher than the resistance for routing flow through the rest of the network. We find that adaptation by such a local, boundary-condition-free measure of redundancy, set by topology alone, yields sparse, loop-rich, low-dissipation networks.

The pronounced chiral asymmetry of biomolecules is a defining feature of living systems, yet the mechanisms by which such asymmetry emerged remain unresolved. Chiral symmetry breaking may have occurred under prebiotic conditions in systems with sufficiently small molecular populations for intrinsic fluctuations to play an important role. Here, we investigate how finite-population fluctuations affect chiral-state selection, the characteristic time for symmetry breaking, and stochastic switching between chiral states in a system driven by autocatalysis and mutual antagonism. We determine how these dynamical processes scale with particle number and derive an analytical expression for the probability of selecting either chiral state as a function of the initial enantiomeric bias. We further calculate the mean selection time for different initial biases and show that, for weak mutual antagonism, fluctuations can induce transitions between opposite chiral states. These results characterize how finite-population fluctuations control both the emergence and stability of molecular chirality in autocatalytic chemical systems.

The effect of mechanical stresses and deformations upon the behavior and properties of cells and tissues is a topic of paramount interest in the field of biophysics. Its understanding is essential for deepening our comprehension of cellular responses to environmental stimuli, with implications in fields ranging from embryogenesis to tumor development. For boosting the understanding in this field, it is important to develop devices for the application of simple deformations in order to probe the cellular response to the basic mechanical transformations. While a number of stretching devices have been developed, in-plane shearing deformation has been little investigated. Here, a new device is introduced that allows the application of homogeneous in-plane shear upon a cell monolayer while imaging the sample with a commercial optical microscope. A description of the faced experimental challenges is reported and a full characterization is presented. It is demonstrated the capability of the device to apply reproducible, controlled and homogeneous shear of both single cells and confluent cellular monolayers.

The imbibition of milk into a biscuit provides an example of a broader class of processes in which a liquid invades a porous material, dissolving components of the solid and thereby modifying its own rheology. This coupling between imbibition, dissolution, and evolving fluid rheology also occurs in food processing, filtration, and reactive porous media. While the resulting changes in fluid viscosity are known to affect imbibition, a quantitative description of how this evolving rheology determines the imbibition dynamics is still lacking. Here, we establish such a description for dissolution-driven fluid thickening. For a capillary tube in the well-mixed limit, we derive a generalized Washburn law showing that the evolving viscosity can be incorporated through a rescaled time given by the accumulated inverse viscosity. This predicts a collapse of the imbibition dynamics for different dissolution rates, which we confirm numerically using coupled free-surface and solute-transport simulations. When diffusion is slower, concentration and viscosity become spatially heterogeneous, resulting in a weaker retardation of the advancing liquid than predicted by the well-mixed theory. We then consider disordered porous media and quantify how dissolution-driven thickening and variations in pore geometry affect the dynamics. Our results extend the classical Washburn description to imbibition with evolving fluid properties and provide a quantitative basis for understanding fluid uptake when interactions with the solid progressively modify the rheology of the invading liquid.

The development of nanostructured antibacterial surfaces has gained significant attention due to the urgent need to combat bacterial colonization and biofouling across food, healthcare, and marine sectors. Among functional metal oxides, zinc oxide (ZnO) nanostructures are particularly promising owing to their rich morphology, favourable physicochemical attributes, and intrinsic bactericidal properties. Here, we utilize a green, eco-friendly Hot Water Treatment (HWT) method to fabricate dense ZnO nanorod arrays on zinc substrates. To overcome the practical limitations and challenges of conventional antibacterial coatings, such as poor operational stability, cytotoxicity, and the risk of antimicrobial resistance, we developed a durable hybrid coating composed of an imidazolium-based ionic liquid stabilized with polydimethylsiloxane. The resulting hybrid-coated nanostructured surfaces exhibit superior antibacterial efficacy against Escherichia coli through a combination of topography-induced mechanical strain and membrane disruption.

Most systems trained by equilibrium propagation or coupled learning are in the same class as multi-layer perceptrons: while they may be universal function approximators, they struggle to learn associative tasks which are easy for transformers. Equilibrium propagation relies on a bidirectional flow of information through the physical system between input and output. The reverse flow is broken by more complex nonlinear systems, such as those performing multiplication which is needed to express dot-product attention. Here we explicitly restore this bidirectional flow by introducing extra circuit elements for this purpose into a double or dyadic circuit. We then use these ideas to construct an attention mechanism exploiting the exponential current law of sub-threshold MOSFET devices. We show that in simulations this circuit can be trained to perform a simple in-context learning task, despite using realistic reverse components which transmit only an approximate gradient. For this training to succeed we need cross-entropy loss (not mean squared error). We generate nudges appropriate for this loss using a novel floating circuit element, based on the same MOSFET sub-circuit used for softmax. We train the resistors in this circuit using the Lion optimiser, which can be implemented using one capacitor per parameter to store local memory of past gradient estimates.

Predicting the structure and response of inhomogeneous polymer electrolytes requires a description of ion correlations that retains molecular-scale accuracy while remaining transferable across spatial scales and geometries. We develop a neural density functional for electrolytes that preserves spatial symmetries, thermodynamic integrability and the Noether identities, with perfect screening recovered in stable, noncritical bulk states. Its nonlinear density dependence captures the concentration-dependent correlations missed by a pair closure, including a crossover from enhanced to suppressed long-wavelength number fluctuations at strong coupling. The functional describes density profiles at an untrained salt concentration and predicts bulk structure factors and the long-wavelength number response. Trained solely on planar density and internal-force profiles from molecular dynamics, the functional predicts ionic structure in larger domains and in two-dimensional external fields. On the same ion data, it is more accurate than three other neural density-functional architectures and keeps its accuracy with a quarter of the training runs, where the errors of the best alternative grow by about two thirds. The spatial transferability provides a necessary foundation for connecting molecular correlations to continuum predictions at larger scales.

Epithelial tissues play a fundamental role in morphogenesis. Mechanically they can be viewed as thin soft materials exhibiting a solid-fluid duality. The Surface Navier-Stokes-Helfrich model accounts for these properties by combining bending and surface hydrodynamics. In order to account for varying cell thickness we incorporate a thickness field in the spirit of a shallow-water equation, but defined on the (self-)evolving surface. This replaces the inextensibility constraint of the two-dimensional fluid and with it the conservation of surface area by an incompressibility constraint of the thin film fluid allowing for changes in surface area. We develop a numerical scheme based on surface finite elements, perform convergence tests, demonstrate the impact on shape evolution of closed surfaces with a constant enclosed volume and discuss implications on modeling morphogenesis.

The strength of a polymer network is orders of magnitude below that of a polymer chain, because the network concentrates high tension in a small fraction of polymer strands. Here we show that the strength of a network can be greatly amplified by recruiting a larger fraction of strands to bear high tension. We develop approaches to fabricate hydrogels of exceptional strength while maintaining low hysteresis. The strength of the hydrogel increases from ~0.05 MPa for a regular network, to ~1 MPa for a highly entangled network, and further to ~10 MPa for a prestretched interpenetrating network. Similar amplifications of strength are achieved for elastomers. Furthermore, experimental data suggest a scaling relation between strength and strand length. This work provides design principles for creating elastic and strong polymer networks.