Papers

For decades, debate has surrounded the electrostatic persistence length (EPL) controlling local polyelectrolyte stiffening, centered on two competing power laws: the linear BJ prediction, $\textbf{l}_\mathrm{e} \sim r_\mathrm{D}$, and the quadratic OSF/KK scaling, $\textbf{l}_\mathrm{e} \sim r_\mathrm{D}^{2}$, where $r_\mathrm{D}$ is the Debye screening length. Building on the asymptotic scaling theory developed in the accompanying paper, we validate a complete diagram of limiting regimes using large-scale coarse-grained Monte Carlo simulations of ideal chains with charged monomers interacting through a screened Coulomb potential. By simulating long chains of up to $N \simeq 10^{4} - 10^{5}$ Kuhn segments, we demonstrate that the electrostatic stiffening of both semiflexible and flexible polyelectrolytes obeys the same quadratic OSF/KK law. We track the exponent $\alpha$ which measures how the chain size $R$ grows with the Debye radius, $R \sim r_\mathrm{D}^{\alpha}$. For both cases, in agreement with the OSF/KK theory, $\alpha$ rises past 3/5 and slowly approaches 1 as the chain length increases, whereas within the BJ theory it can never exceed 2/5. We further find that three common size measures, the end-to-end distance $R_\mathrm{ee}$, radius of gyration $R_\mathrm{g}$, and hydrodynamic radius $R_\mathrm{h}$, reach this asymptotic behavior at progressively increasing chain lengths. Consequently, for any real polyelectrolyte, the apparent exponents obey $\alpha_\mathrm{ee} \geq \alpha_\mathrm{g} \geq \alpha_\mathrm{h}$, making $R_\mathrm{g}$ a sharper experimental probe than $R_\mathrm{h}$. Finally, we re-analyze the available experimental data and show that they rule out the linear BJ law and support the quadratic KK scaling, thereby resolving contradictions that stem from mistaking the apparent slopes measured for short chains for the true asymptotic exponent.

The problem of single-chain conformations of polyelectrolytes in salt-added dilute solutions, and the associated concept of the electrostatic persistence length $\textbf{l}_\mathrm{e}$, has remained unresolved for decades. To address this challenge, we develop a comprehensive scaling theory and corroborate it with simulations. A unified scaling diagram is constructed that encompasses both flexible and intrinsically semiflexible/stiff polyelectrolytes. Nine distinct scaling regimes are identified, each characterized by different conformational statistics. The concept of the Gaussian electrostatic blob $\xi_\mathrm{e}$ used in the description of flexible polyelectrolytes is extended to the semiflexible case, and the new characteristic length $\xi_\mathrm{e}^{|}$ referred to as the rodlike electrostatic blob is introduced. This enables delineating the important crossovers and showcasing an analogy between semiflexible and flexible chains. The concept of electrostatic excluded volume is also reconsidered and generalized. Upon increasing the salt concentration, i.e., decreasing the Debye length $r_\mathrm{D}$, chain conformations evolve from (i) rodlike stretches to (ii) Gaussian and then (iii) swollen coils with local electrostatic stiffening characterized by $\textbf{l}_\mathrm{e}$, followed by (iv) swollen coils without stiffening but with electrostatic exclude volume, and finally to (v) quasi-neutral Gaussian coils. In regimes of type (ii) and (iii), the electrostatic persistence length scales quadratically with $r_\mathrm{D}$, in agreement with the OSF and KK predictions for semiflexible and flexible chains, respectively: $\textbf{l}_\mathrm{\mathrm{OSF}} \simeq r_\mathrm{D}^{2}/\xi_\mathrm{e}^{|}$ and $\textbf{l}_\mathrm{\mathrm{KK}} \simeq r_\mathrm{D}^{2}/\xi_\mathrm{e}$. These asymptotic scalings are confirmed by coarse-grained simulations presented in the accompanying article.

Multicomponent liquids can phase separate upon heating, exhibiting a lower critical solution temperature (LCST). Moreover, a narrow class of materials can undergo disordering transition upon further heating, yielding closed-loop phase diagrams. Previously, it was shown that LCST or closed-loop phase behavior can appear in the models of liquids in which components are asymmetric or interaction potentials have a specifically designed attraction. Here, we show theoretically that LCST and closed-loop phase behavior can occur in a significantly wider set of models. In particular, we found that these phenomena can be exhibited by the simplest and widely used coarse-grained (CG) models of any multicomponent liquid in which species are fully symmetric and where all particles interact via an arbitrary repulsive potential. In addition, we discovered that LCST and closed-loop phase behavior in these models emerges merely due to the basic property of CG liquid models, namely, the appearance of strong monomer-monomer positional correlations at low monomer number density $\rho$. We simulated the models of fully symmetric binary blends and diblock copolymer melts where nonbonded monomers interacted via a generic T-independent purely repulsive harmonic potential. As predicted, LCST and closed-loop phase behavior emerged at $\rho$ sufficiently low to cause strong monomer-monomer correlations leading to a strong T-dependence of the effective coordination number, which, in turn, induced a nonmonotonic T-dependence of the Flory-Huggins parameter. To summarize, we discovered that the simplest fully symmetric non-associating CG models of multicomponent liquids can exhibit complex temperature response due to a mechanism stemming from the basic nature of any CG liquid model. This mechanism might contribute to the emergence of LCST and closed-loop phase behavior in many existing polymer materials.

The dramatic slowing down of structural relaxation in supercooled liquids is accompanied by the emergence of dynamic heterogeneity. A monotonically increasing dynamical correlation length, measured at the $\alpha$-timescale, is one of the remarkable features of this phenomenon. Here we show that this picture is incomplete: the dynamical correlation length measured in the $\beta$-relaxation regime exhibits a striking non-monotonic temperature dependence, reaching a maximum near the mode-coupling crossover temperature $T_c$ and decreasing upon further cooling, even as local dynamical fluctuations continue to intensify. This behavior suggests a crossover from spatially extended, maximally cooperative motion near $T_c$ to increasingly compact and localized relaxation events below it. We demonstrate that this evolution is quantitatively captured by stochastic beta-relaxation theory, an extension of mode-coupling theory beyond mean-field that explicitly predicts an avoided dynamical transition in finite dimensions. Our results provide the first direct spatial evidence in favor of the avoided-transition picture of the mode-coupling crossover, and establish the peak of the $\beta$-regime correlation length as a robust indicator of the mode-coupling crossover.

We show that living microswimmers mechanically reinforce dense emulsions. Castor-oil-in-water emulsions laden with the microalga Chlamydomonas reinhardtii are compared in three states: without algae, with immobilized algae, and with motile algae, over a broad range of oil fractions spanning the jamming transition. Oscillatory rheology reveals that motile algae systematically increase the yield stress, by up to a factor of two, whereas immobilized cells at the same concentration leave it essentially unchanged. The reinforcement thus originates from activity rather than from the mere presence of inclusions. Single-cell tracking shows that the droplet network confines the swimmers in pores that shrink as the oil fraction increases, and confinement is known to amplify the propulsion force of C. reinhardtii. A simple estimate based on this confinement-enhanced force accounts for the measured excess yield stress and indicates an effective, activity-induced depletion-like attraction between the passive droplets. These results identify a feedback loop: the microstructure confines the swimmers, confinement amplifies the forces they exert, and these forces stiffen the microstructure. Active emulsions thus emerge as a model platform for programming the mechanics of disordered soft solids through activity.

Artificial intelligence (AI) is redefining the landscape of polymer science. Although numerous AI applications have been introduced in this field, the roles of polymer encoding strategies and different applications of AI models in polymer design remain insufficiently understood. Here, we build upon the foundation of polymer databases to critically examine the performance and applicability of current encoding strategies across different use cases. We then focus on two major AI application domains, property prediction and inverse design, to evaluate the strengths, weaknesses and suitable scenarios for various model architectures. Finally, we emphasize the significance of the online platforms for promoting data accessibility and accelerating the migration from experience-based discovery toward AI-driven innovation in polymer science and engineering. Through these discussions, we aim to provide practical guidance for future research and development in AI-assisted polymer design.

Polar liquid crystals - fluid phases in which molecular dipoles organise into ferro- or antiferroelectric states - present a highly constrained molecular design space, where small changes can entirely suppress polar organisation. Here we demonstrate, contrary to expectations, that installation of a methyl group in the 5-position of a 1,3-dioxane ring substantially increases the onset temperature of polar order. Remarkably, this effect proves transferable across several liquid-crystal families, including examples where the methylated derivative exhibits a polar phase whereas its parent compound does not. Across 8 matched pairs, we find that a 5-methyl group can enhance the onset temperature of polar order in 1,3-dioxane materials by up to 120 {\deg}C. These findings establish a simple and general molecular design strategy for enhancing and enabling spontaneous dipolar ordering in liquid crystals.

The Trachenko-Zaccone equation provides a compact nonlinear dynamical framework for describing non-Debye relaxation in disordered condensed matter. Originally developed to rationalize stretched- and compressed-exponential relaxation in liquids and glasses from the dynamics of interacting local relaxation events, the same equation has subsequently appeared in broader contexts, including polymer relaxation and nonlinear models of global population dynamics. This review retraces the conceptual development of the equation, with particular emphasis on its physical origin in Kostya Trachenko's feed-forward interaction mechanism, its mathematical structure, and its possible generalizations. We also include personal recollections of the work with Kostya Trachenko at Queen Mary University of London in August 2019, during which the equation emerged in essentially its present form. After reviewing applications to stress relaxation, glassy materials, polymer relaxation and population dynamics, we discuss future directions including time-dependent feedback parameters, coupled order parameters, heterogeneous and spatially resolved formulations, flux terms, network versions and stochastic extensions. The central theme of the review is that the Trachenko-Zaccone equation should be viewed not only as a model of glassy relaxation, but as a general nonlinear feedback equation with potential applications across complex systems.

Thermal fluctuations can drive an interface when bulk fluctuation amplitudes differ between two phases. We study how shear flow modifies this mechanism using a stochastic non-conserved order- parameter model. For a planar interface parallel to the imposed shear, we derive its propagation velocity in the weak-noise and small-bias regime. The driving force comprises a shear-modified entropic contribution determined by bulk fluctuation spectra and a non-equilibrium contribution arising from the breaking of time-reversal symmetry. At low shear rates, the entropic term provides the dominant contribution to the velocity of an interface near the zero-velocity plane of the shear flow, and the shear-induced change in the velocity scales as the 4/3 power of the shear rate. At high shear rates, two-dimensional simulations show that the measured velocity is largely accounted for by the modified bulk entropic contribution.

Mobility links an applied force to the resulting motion and is generally positive near equilibrium. Far from equilibrium, however, internal energy input can reverse this response. Here we show that information feedback provides a distinct route to negative mobility. We consider an overdamped dimer consisting of a run-and-tumble particle coupled by a spring to a passive Brownian particle. Information enters through periodic measurements of the relative displacement, which are processed to reset the active polarity. Although the unloaded dimer has no net drift, an applied force biases its internal configuration, and an information-mechanical feedback amplifies and converts this bias into active propulsion against the force, producing negative mobility. We develop an analytical theory that captures this mechanism and yields a feedback-gain criterion for response reversal. Including the information-processing cost reveals a tradeoff: rapid feedback enhances reverse transport but incurs a growing informational cost, yielding an optimal finite feedback rate for information-inclusive efficiency. These results show that information can reshape nonequilibrium response by controlling how internally supplied energy is converted into macroscopic transport.

The mechanical properties of polymer glasses are often critical in determining the best material for a particular application. Extremely stiff materials (high modulus) may be important for some applications while avoiding catastrophic failure due to fracture (high toughness) may be more important for others. The mechanical properties of a polymer glass will depend upon both molecular structure and many experimental variables, including temperature and the mode of deformation (tension, compression, or shear). In this chapter we discuss the mechanical response of polymer glasses from a molecular perspective. In particular, we consider how deformation changes the rate at which polymer segments rearrange and how this in turn influences the mechanical response of the material. It will be shown that this focus on the changes in dynamics provides an understanding of many important features of polymer glass deformation. Of course, it is also true that the structure of a polymer glass must be altered by nonlinear deformation. Although not a major focus, we will make some comments about these structural changes at the end of this chapter.

Glasses of a model smectic liquid crystal-forming molecule, itraconazole, were prepared by vapor deposition onto substrates with temperatures ranging from Tsubstrate = 0.78 Tg to 1.02 Tg, where Tg = 330 K is the glass transition temperature. The films were characterized using x-ray scattering techniques. For Tsubstrate near and below Tg, glasses with layered smectic-like structures can be prepared and the layer spacing can be tuned by 16% through choice of Tsubstrate. Remarkably, glasses prepared with Tsubstrate above Tg exhibit much higher structural organization than a thermally annealed film. These results are explained by a mechanism based upon preferred molecular orientation and enhanced molecular motion at the free surface, indicating that molecular organization in the glass is independent of the anchoring preferred at the substrate. These results suggest new strategies of optimizing molecular packing within active layers of organic electronic and optoelectronic devices.

Optical probe reorientation measurements were performed to monitor changes in segmental dynamics resulting from the nonlinear deformation of a polymer glass. Segmental dynamics were monitored in a poly(methyl methacrylate) glass near Tg before and after a series of reversing deformations in which the sample was extended at constant strain rate and then allowed to retract back to zero stress at constant strain rate. Evidence of a rejuvenation mechanism, as quantified by a departure of the segmental dynamics from the quiescent aging dynamics after the reversing deformation, is observed for deformations which reach 60% of the yield strain or greater. By this measure, a saturation of the rejuvenation mechanism is not observed until at least five times the yield strain. For comparison, purely mechanical measurements of rejuvenation, based upon the reduction of the yield stress in a subsequent deformation, were also performed. These purely mechanical experiments show broad qualitative agreement with the probe reorientation experiments, but quantitatively differ in the pre-yield regime. The results are discussed in the context of recent theoretical approaches and simulations which provide a molecular-level description of polymer glass deformation.

A tangle of line defects coarsens as the mean spacing $L$ between neighbouring lines grows. In a viscous medium the motion is overdamped, and the velocity-dependent one-scale (VOS) model predicts a late-time attractor $L^{2}=\mathcal{A}\,\ell_d\,t$, with $\ell_d=T/\Gamma$ the ratio of line tension to drag and $\mathcal{A}$ a dimensionless amplitude. The growth law $L\propto t^{1/2}$ holds for every value of the three model parameters—the momentum parameter $k\le1$, the sink coefficient $\tilde{c}$, and the curvature ratio $\lambda\equiv R/L$—since all three enter only through $\mathcal{A}=\kappa(\kappa+\tilde{c})$, $\kappa\equiv k/\lambda$. Thus, the exponent constrains none of them, and the amplitude is the only quantity a density history can deliver. We measure it from the disclination data of Chuang, Turok and Yurke on a nematic liquid crystal, whose companion measurement of loop collapse fixes $\ell_d$ on the same samples, canceling the 5CB material constants. Treating the unmatched per-quench $\ell_d$ as a nuisance parameter with a Gaussian prior and marginalizing it analytically, we obtain $\mathcal{A}=10.0^{+1.3}_{-1.1}$ from the three quenches in the $234~\mu$m cell; the fourth, the only one in the thinner $158~\mu$m cell, gives $\mathcal{A}=3.0^{+0.8}_{-0.6}$ and is fitted separately. Converting either into $\tilde{c}$ requires $\lambda$, which these data do not determine, so the result is a curve, $\tilde{c}(\lambda)=\mathcal{A}\lambda/k-k/\lambda$, not a number. At $\lambda=1$ with $k\le1$ they give $\tilde{c}\ge9.0$ and $\tilde{c}\ge2.0$, against $\tilde{c}=0.23$–$0.57$ from relativistic $U(1)$ simulations, values reached only at $\lambda\simeq0.33$–$0.35$ and $0.62$–$0.68$. We know of no VOS calibration for a cosmological $\mathbb{Z}_2$ network, so we cannot attribute the excess to topology, and we list what a repeat experiment must measure.

Growth may cease at a target size or continue throughout life: the determinate and indeterminate phenotypes. We develop an active viscoelastic continuum model of a tissue growing along one axis, in which cell division and death generate active stresses. We find two asymptotic states: one in which the tissue reaches a relative size fixed by its material parameters, and one in which it elongates linearly without bound. Which state is realised is set by the ratio of active stress to elastic modulus. The transition originates in a bound on the elastic stress the tissue can support: a sufficiently large activity can never be balanced. In a tissue made of parts with different material properties, the growing phase settles into fixed length proportions, set by the mechanical impedances of the parts rather than inherited; matching impedances to initial lengths preserves the proportions the tissue began with. Determinate, indeterminate and proportionate growth thus appear as regimes of one continuum mechanical framework.

Recent years have seen substantial progress in elucidating the statistical physics of the vibrational properties of structural glasses. Although many real-world glasses relevant for science and technology are molecular, the majority of computational studies concerning the mechanical and vibrational properties of structural glasses employ simple atomistic glass-forming models. Thus, the effects of stiff molecular constraints on mechanical and vibrational glass physics remain largely unexplored. In this work, we directly compare the properties of a molecular computer glass with those of an atomistic model featuring the same inter-molecular interaction potential, and created using the same formation protocol. We find that the molecular glass features a higher degree of mechanical disorder, with larger mesoscopic correlation lengths, while at the same time it hosts a lower number of soft, quasilocalized vibrations per atom – compared to the atomistic glass. We rationalize these differences by accounting for the reduction in the effective number of degrees of freedom induced by the stiff molecular constraints. We additionally find that nonlinear plastic modes — that carry plastic deformation in driven glassy solids — couple much more strongly to volumetric strains in the molecular glass. Future research directions are discussed.

The elastic modulus of porous materials is commonly described using power-law scaling relations with relative density, where the scaling exponent is often interpreted in terms of the underlying deformation mechanism. However, in highly disordered porous networks, changes in density are generally accompanied by changes in network topology, which can substantially modify the apparent scaling behavior. In this paper, we propose a topology-informed framework that separates the intrinsic mechanical contribution from the effects of network structure. Three representative topological descriptors are considered: the mean coordination number, the fraction of the load-bearing backbone, and the tortuosity of the load paths. For each case, the corresponding density-dependent contribution to the apparent modulus-scaling exponent is derived and analyzed. The results show that variations in connectivity, mechanical participation of the solid phase, and load-path efficiency can all lead to apparent scaling exponents exceeding the intrinsic exponent associated with the local deformation mechanism. These effects are particularly pronounced at low relative densities, where network topology evolves most strongly. The framework provides a physically interpretable basis for understanding anomalous modulus-density scaling in disordered porous materials and highlights the need to consider topology explicitly alongside relative density.

Casein micelles, one of the most studied natural association colloids, are supramolecular assemblies of caseins and colloidal calcium phosphate that constitute the fundamental building blocks of dairy matrices. Despite extensive investigation, the internal structure of casein micelles remains debated. While $\kappa$-casein is known to ensure colloidal stability of casein micelle suspension, the spatial organization of casein fractions and salts is still unresolved, and several structural models coexist. Small-angle scattering is a method of choice to probe biological colloids in situ, yet interpretation of scattering data remains challenging due to the hierarchical nature of casein micelles. Here we address this issue by probing micelle structure during enzymatic gelation induced by chymosin, which cleaves $\kappa$-casein and triggers aggregation. Using time-resolved SAXS, we probe structural changes throughout the sol-gel transition over length scales from 3 nm to 3 $\mu$m. First, we show that enzyme-driven aggregation, counterintuitively, reveals information about the internal organization of casein micelles: the reduction of specific surface area during gelation unmasks a high-q structural peak previously observed under contrast-matching conditions. Second, we report that $\kappa$-casein cleavage leads to a gradual disappearance of the structural feature at intermediate scales. Analysis of the disappearance kinetics and comparison to structural models reveal that $\kappa$-casein cleavage induces a progressive relaxation of the colloidal porous substructure, providing direct evidence for its contribution to micellar organization. More broadly, these results demonstrate that the gelation process provides unique access to the internal structure of biological colloids and offers new perspectives for interpreting scattering data in complex soft-matter systems.

Time-varying interaction delays are ubiquitous in active matter, yet their collective effects remain largely unexplored. We show that active particles with internal dynamics and Vicsek-like delayed alignment exhibit intermittent flocking, characterized by long episodes of coherent motion interrupted by brief disordering events. This collective behavior arises from laminar chaos, a form of chaotic dynamics unique to systems with time-varying delays. Changing only the delay parameters qualitatively reshapes collective motion, producing fractal changes in global flocking order. Our results establish the temporal structure of interaction delays as a new control parameter for active matter.

Thermoresponsive microgels undergo large reversible size changes as temperature alters solvent quality. Predicting their nonequilibrium swelling and deswelling kinetics is challenging because polymer volume fraction, mechanical response, and solvent transport evolve during large volume changes. Here we develop a solvent-flux theory for the dynamics of a spherical microgel. The radius-change rate is driven by the osmotic-pressure imbalance across the particle boundary and resisted by water transport through the polymer network, yielding a nonlinear equation for the global swelling coordinate and a state- and temperature-dependent swelling diffusion coefficient, $D_{\mathrm{SW}}(\phi,T)$. In the linear-response regime, the theory recovers Tanaka–Fillmore exponential relaxation and the scaling $\tau_\mathrm{SW}\sim R_{\mathrm{eq}}^2\gamma/K_{\mathrm{eq}}$, while providing a microscopic interpretation of the polymer–solvent friction coefficient $\gamma$. Beyond this limit, the model retains quadratic size scaling while accounting for state-dependent transport and mechanics. For pNIPAM microgels, it predicts asymmetric pathways, with deswelling faster than swelling over the same temperature interval. Finite thermalization produces a crossover from a microgel-controlled to a thermalization-controlled regime, in which the apparent relaxation time grows linearly with the external thermalization time and hysteresis-like loops emerge in the radius–temperature plane. A stochastic extension based on the Smoluchowski equation predicts transient broadening of the radius distribution during collapse, with maximal fluctuations in the volume-transition region. The theory links solvent transport, nonlinear swelling dynamics, thermalization effects, and nonequilibrium size fluctuations in responsive microgels.

Molecular docking is a vital computational task in drug discovery, wherein the objective is to efficiently identify optimal binding poses between a ligand and a target receptor protein. Due to the combinatorial explosion of possible binding configurations, docking of large and flexible molecules remains a computationally intensive problem, especially at scale. Early studies have revealed that the molecular docking can be re-cast as a maximum vertex-weighted clique problem (MVWCP) problem on a compatibility graph to be solved classically. In this work, we proposed a hybrid quantum-classical approach for molecular docking leveraging the MVWCP formalism with a variational full-basis encoding (FBE) strategy, which enables efficient encoding of classical binary variables with Bloch sphere vectors. We further prove that a global minimizer of the FBE objective can always be chosen to be a pure product state, thereby providing a rigorous justification for its optimization using a unitary variational circuit. The molecular docking problem is first mapped to a cost Hamiltonian that is minimized within a variational framework, optimized via a randomized imaginary time evolution (ITE)-inspired warm start, and gradient-based techniques. Finally, we also executed the circuit on an IBM quantum computer, underlying the feasibility and of quantum-assisted optimization for structure-based drug design and point towards the broader utility of advanced encoding techniques in quantum optimization for computational biology.

The self-assembly of associative triblock copolymers composed of a central hydrophilic elastin-like polypeptide (ELP) block and short fatty acid end groups (C16) was investigated in aqueous solution. In one system, the ELP contains 80 pentapeptide units (C16-80-C16), whereas in the other two C16-ELP40 chains were oxidatively coupled through their terminal cysteine residues to form a central disulfide bond, yielding C16-(40)2-C16. Despite their nearly identical molecular weights and compositions, the two polymers exhibit markedly different self-assembly behaviors. C16-80-C16 forms large hydrophobic aggregates that remain kinetically trapped and do not develop a dynamically connected network. In contrast, C16-(40)2-C16 forms very small associative nodes with an aggregation number of only $\sim$3 chains. These nodes coexist with larger clusters and become dynamically interconnected at higher concentrations, leading to transparent hydrogels at concentrations as low as 2.5 wt %. Oscillatory rheology reveals a transient Maxwell network governed by a single relaxation process associated with the reversible association of the C16 end groups. SAXS, light scattering, cryo-TEM, and molecular modeling consistently support a model in which the central disulfide junction promotes transient network formation.

Open-porous materials exhibit pronounced compressibility, nonlinear densification, and power-law scaling of stiffness with density. In this work, we propose a thermodynamically consistent compressible neo-Hookean constitutive model for open-porous solids in which porosity serves as the primary governing variable. The strain-energy density is formulated to couple distortional elasticity of the solid skeleton with a volumetric response governed by deformation-induced porosity evolution, including a bounded representation of pore collapse. The formulation introduces a minimal set of parameters, namely the initial porosity, intrinsic skeleton moduli, and a scalar parameter controlling the onset of densification. A key feature of the model is a modified volumetric term in which the response is normalised by the current porosity, ensuring a physically consistent transition from a porous to a densified state without artificial stiffening. In the small-strain limit, the model recovers classical linear elasticity with effective moduli that may be chosen either from homogenisation bounds, such as the Hashin-Shtrikman estimates, or from Gibson-Ashby-type power-law scaling to capture topology-dependent behaviour. At finite strains, the formulation captures the characteristic nonlinear stiffening and convex stress-stretch response associated with progressive pore collapse. The proposed framework thus provides a compact, flexible, and extensible constitutive description that unifies effective-medium consistency with experimentally observed scaling behaviour, and is well suited for finite element implementation and multiscale modelling of highly compressible open-porous materials. The model is finally validated against available experimental data.

Lipid-coated microscale hydrogels provide confined, hydrated environments in which polymer phase behavior can differ markedly from that in bulk. Here, we investigate the crystallization pathway of poly(ethylene glycol) (PEG) encapsulated in lipid-coated agarose microgels. Surprisingly, polarized-light microscopy reveals birefringence in the microgels under conditions where PEG remains non-crystalline in the corresponding bulk solution. The birefringence disappears upon heating and spontaneously reappears after further cooling or upon local mechanical stimulation. Infrared microspectroscopy demonstrates that the birefringent microgels contain crystalline PEG, whereas non-birefringent microgels contain PEG in an amorphous-like state, indicating the existence of a metastable precursor prior to crystallization. Furthermore, cooling below the phase-separation temperature produces PEG-rich domains preferentially near the membrane, suggesting that membrane wetting governs the spatial distribution of PEG before crystallization. Together, these results indicate that membrane confinement and agarose gelation cooperatively alter the local hydration environment of PEG, thereby stabilizing an amorphous-like precursor and redirecting the subsequent crystallization pathway. Our findings identify the coupling of membrane wetting and gelation as a key factor governing PEG crystallization in confined soft materials.

How many memories can a local learning rule protect, and what fixes the number? Physical learning rules train resistive networks through local measurements, and the standard rules conserve a mass-like function of the conductances, a law whose general form was posed as an open problem with the expectation that no useful solution theory exists. We answer it, in the direction the expectation ran against, and the answer is a capacity theory. The Tellegen identity behind the conserved mass localizes: the feedback state is pinned to zero at the inputs, so every sector of the circuit that the input electrodes separate carries its own private conserved mass, by an argument consuming only Kirchhoff's law and that boundary condition, hence valid for arbitrary nonlinear branch laws. The number of independent sector masses is a topological property of the circuit, bounded by the number of output electrodes. An untrainable element can destroy the mass of its own sector and of no other, which says where fixed nonlinear components may be placed; per-edge learning rates select which functionals are conserved and never how many; and adjoint coupled learning, which clamps its outputs before measuring, drains every output-carrying sector at a rate set by the squared output multipliers. In linear circuits we then classify the identities: exact rational computation over all 502 circuits on up to five vertices with two inputs and one or two outputs, and hundreds of larger ones, yields a closed-form count, proved on that family and conjectured in general, with the sector statement a theorem at every circuit size. The anticipated series-parallel polynomial laws appear as exactly the series-class cubic differences, and the number the opening question asks for is a budget: one designed mass per sector, the differences the topology donates, and one broadcast scalar for each protected functional beyond.

Even for tension-dominated fracture propagation, porous materials may deform plastically adjacent to the propagating fracture. As is common for porous materials, existing phase-field models typically employ a non-associative flow rule for plasticity, and a Helmholtz free energy based on strain and fluid pressure. This work revisits the hydromechanically coupled formulation of the phase-field model for fracture in poro-elastoplastic media by analyzing the strength surface and fracture driving force. Our analyses show that these common choices of flow rule and free energy will lead to a discontinuous strength surface across the tension-compression transition. A non-associative flow rule introduces a jump at the strength surface, while treating fluid pressure-rather than fluid content-as the independent variable in the Helmholtz free energy omits a coupling term from the phase-field driving force, also breaking continuity. Incorporating an associative Drucker-Prager flow rule and this omitted coupling term ensures a continuous strength surface and the accurate fracture driving force. The proposed model exhibits improved accuracy in hydromechanical responses when compared against the analytical solution of the Kristianovich-Geertsma-de Klerk hydraulic fracturing benchmark. Numerical simulations of hydraulic fracturing and biaxial compression in poro-elastoplastic media show that the model can reproduce both shear-dominated fractures induced by mechanical disturbance and tension-dominated fractures driven by fluid injection in saturated porous media.

In layered materials, spacing and stacking-order extent are usually locked. Here we show that in swollen suspensions of stiff, charged nanosheets they decouple, and that this defines a distinct regime, apart from the crystalline- and Wigner-swelling regimes such systems usually occupy. The mean spacing stays sharp and salinity-tunable while scattering-weighted stacking spans only two to three layers. We demonstrate this in a near-perfect model material, so the behaviour is intrinsic, not defect-driven. X-ray and neutron scattering, sedimentation and a Donnan analysis show the spacing is held by a parameter-free osmotic restoring slope below one pascal per nanometre. Because the slope is so weak, the spacing sits at equilibrium while faults relax slowly, a quenched metastable registry whose ageing-like relaxation of low-dimensional periodic order has not, to our knowledge, been realised before. The same decoupling is expected across stiff, swollen nanosheets, from clays to oxide nanosheets and graphene oxide.

We experimentally and theoretically study the hydrodynamic analog of the classical brachistochrone problem: the time- and energy-minimizing paths for a spherical particle rolling down an incline within a viscous fluid. We show that in the presence of viscous dissipation, the paths of minima diverge from the classical cycloid, into curves of opposing curvature for time and energy, and are characterized by an effective dimensionless parameter, $St_p$, representing the ratio of the particle's viscous response time scale to its gravitational time scale. Using a generalized variational framework, we show that the fastest path reduces to {nearly straight ramps}, however, beginning and terminating in localized cycloids of curvature, $\kappa_c \sim St_p^{-2}$. Remarkably, the path of fastest descent on a given energy budget requires navigating a non-monotonic path ({\it“S-shaped”}) with an interior point of inflection. Our findings reveal a unification of temporal and energetic optimality for transport through dissipative media, and expand the celebrated brachistochrone solutions to the hydrodynamic regime.

We numerically examine the impacts of particle-wall alignment interactions on active species diffusion through a structureless narrow two-dimensional channel. We consider particle-wall interaction to depend on the self-propulsion velocity direction whereby some specific particle's alignments with respect to the boundary walls are stabilized most. Further, the alignment interaction is meaningful as long as particles are close to the confining boundaries. Unbiased diffusion of active particles for various possible stable velocity alignments against the walls has been examined. We show that for the most stable configuration leading to self-propulsion velocity direction perpendicular to the wall, diffusivity becomes inversely proportional to the square of alignment interaction torque. On the other hand, when the self-propulsion velocity direction making an acute angle to the channel walls is the most stable configuration, diffusion exponentially grows with strengthening alignment interaction. Hence, particle-wall interaction plays a pivotal role in the transport control of active particles through narrow channels. Moreover, the impacts of the alignment interactions on diffusion largely depend on the particle's self-propulsion properties and its chirality. Our simulation results can potentially be used to understand unbiased diffusion of artificial or living micro/nano-objects (such as virus, bacteria, Janus particles, etc.) though narrow confined structures.

Understanding how life-like behaviors can emerge from simple molecular assemblies and primitive compartments remains a central challenge in origins-of-life research. Synthetic minimal cells provide a bottom-up platform for investigating, from scratch, the minimal physicochemical principles underlying compartment growth, reproduction, and evolution. Previously, we developed a vesicle/polymer-based compartment system in which the vesicle membranes template the formation of a catalytic polymer. This polymer promotes selective incorporation of amphiphiles into the vesicle membrane, driving vesicle growth while maintaining the compositional identity and enabling spontaneous deformation and division over several generations. Here, we report about experiments in which we advanced this system beyond reproduction by systematically constructing eight synthetic minimal cell variants from combinations of two template vesicles, two catalytic polymers, and two supplied amphiphiles. The variants exhibited distinct, composition-dependent vesicle growth responses, ranging from pronounced growth to suppressed growth or vesicle shrinkage. These growth responses were described by the Hill kinetics and characterized by three parameters, revealing a multi-dimensional fitness landscape shaped by environmental conditions, in which the relative advantage of each variant depends on both composition and amphiphile availability. This framework links molecular recognition, compositional inheritance, and differential growth, providing a physicochemical route toward evolvable synthetic minimal cells.