Auxetic polymer networks: The role of crosslinking, density, and disorder.

Low-crosslinked polymer networks have recently been found to behave auxetically when subjected to small tensions, that is, their Poisson's ratio ν becomes negative. In addition, for specific state points, numerical simulations revealed that diamond-like networks reach the limit of mechanical stability, exhibiting values of ν = -1, a condition that we define as hyper-auxeticity. This behavior is interesting per se for its consequences in materials science but is also appealing for fundamental physics because the mechanical instability is accompanied by evidence of criticality. In this work, we deepen our understanding of this phenomenon by performing a large set of equilibrium and stress-strain simulations in combination with phenomenological elasticity theory. The two approaches are found to be in good agreement, confirming the above results. We also extend our investigations to disordered polymer networks and find that the hyper-auxetic behavior also holds in this case, still manifesting a similar critical-like behavior as in the diamond one. Finally, we highlight the role of the number density, which is found to be a relevant control parameter determining the elastic properties of the system. The validity of the results under disordered conditions paves the way for an experimental investigation of this phenomenon in real systems, such as hydrogels.

Structure and elasticity of model disordered, polydisperse, and defect-free polymer networks.

The elasticity of disordered and polydisperse polymer networks is a fundamental problem of soft matter physics that is still open. Here, we self-assemble polymer networks via simulations of a mixture of bivalent and tri- or tetravalent patchy particles, which result in an exponential strand length distribution analogous to that of experimental randomly cross-linked systems. After assembly, the network connectivity and topology are frozen and the resulting system is characterized. We find that the fractal structure of the network depends on the number density at which the assembly has been carried out, but that systems with the same mean valence and same assembly density have the same structural properties. Moreover, we compute the long-time limit of the mean-squared displacement, also known as the (squared) localization length, of the cross-links and of the middle monomers of the strands, showing that the dynamics of long strands is well described by the tube model. Finally, we find a relation connecting these two localization lengths at high density and connect the cross-link localization length to the shear modulus of the system.

Onset of criticality in hyper-auxetic polymer networks.

Against common sense, auxetic materials expand or contract perpendicularly when stretched or compressed, respectively, by uniaxial strain, being characterized by a negative Poisson's ratio ν. The amount of deformation in response to the applied force can be at most equal to the imposed one, so that ν = - 1 is the lowest bound for the mechanical stability of solids, a condition here defined as "hyper-auxeticity". In this work, we numerically show that ultra-low-crosslinked polymer networks under tension display hyper-auxetic behavior at a finite crosslinker concentration. At this point, the nearby mechanical instability triggers the onset of a critical-like transition between two states of different densities. This phenomenon displays similar features as well as important differences with respect to gas-liquid phase separation. Since our model is able to faithfully describe real-world hydrogels, the present results can be readily tested in laboratory experiments, paving the way to explore this unconventional phase behavior.

Two-step deswelling in the Volume Phase Transition of thermoresponsive microgels.

Thermoresponsive microgels are one of the most investigated types of soft colloids, thanks to their ability to undergo a Volume Phase Transition (VPT) close to ambient temperature. However, this fundamental phenomenon still lacks a detailed microscopic understanding, particularly regarding the presence and the role of charges in the deswelling process. This is particularly important for the widely used poly(N-isopropylacrylamide)-based microgels, where the constituent monomers are neutral but charged groups arise due to the initiator molecules used in the synthesis. Here, we address this point combining experiments with state-of-the-art simulations to show that the microgel collapse does not happen in a homogeneous fashion, but through a two-step mechanism, entirely attributable to electrostatic effects. The signature of this phenomenon is the emergence of a minimum in the ratio between gyration and hydrodynamic radii at the VPT. Thanks to simulations of microgels with different cross-linker concentrations, charge contents, and charge distributions, we provide evidence that peripheral charges arising from the synthesis are responsible for this behavior and we further build a universal master curve able to predict the two-step deswelling. Our results have direct relevance on fundamental soft condensed matter science and on applications where microgels are involved, ranging from materials to biomedical technologies.

Effect of Chain Polydispersity on the Elasticity of Disordered Polymer Networks.

Due to their unique structural and mechanical properties, randomly cross-linked polymer networks play an important role in many different fields, ranging from cellular biology to industrial processes. In order to elucidate how these properties are controlled by the physical details of the network (e.g., chain-length and end-to-end distributions), we generate disordered phantom networks with different cross-linker concentrations C and initial densities ρinit and evaluate their elastic properties. We find that the shear modulus computed at the same strand concentration for networks with the same C, which determines the number of chains and the chain-length distribution, depends strongly on the preparation protocol of the network, here controlled by ρinit. We rationalize this dependence by employing a generic stress-strain relation for polymer networks that does not rely on the specific form of the polymer end-to-end distance distribution. We find that the shear modulus of the networks is a nonmonotonic function of the density of elastically active strands, and that this behavior has a purely entropic origin. Our results show that if short chains are abundant, as it is always the case for randomly cross-linked polymer networks, the knowledge of the exact chain conformation distribution is essential for correctly predicting the elastic properties. Finally, we apply our theoretical approach to literature experimental data, qualitatively confirming our interpretations.

Dynamical properties of different models of elastic polymer rings: Confirming the link between deformation and fragility.

We report extensive numerical simulations of different models of 2D polymer rings with internal elasticity. We monitor the dynamical behavior of the rings as a function of the packing fraction to address the effects of particle deformation on the collective response of the system. In particular, we compare three different models: (i) a recently investigated model [N. Gnan and E. Zaccarelli, Nat. Phys. 15, 683 (2019)] where an inner Hertzian field providing the internal elasticity acts on the monomers of the ring, (ii) the same model where the effect of such a field on the center of mass is balanced by opposite forces, and (iii) a semi-flexible model where an angular potential between adjacent monomers induces strong particle deformations. By analyzing the dynamics of the three models, we find that in all cases, there exists a direct link between the system fragility and particle asphericity. Among the three, only the first model displays anomalous dynamics in the form of a super-diffusive behavior of the mean-squared displacement and of a compressed exponential relaxation of the density auto-correlation function. We show that this is due to the combination of internal elasticity and the out-of-equilibrium force self-generated by each ring, both of which are necessary ingredients to induce such a peculiar behavior often observed in experiments of colloidal gels. These findings reinforce the role of particle deformation, connected to internal elasticity, in driving the dynamical response of dense soft particles.

Charge affinity and solvent effects in numerical simulations of ionic microgels.

Ionic microgel particles are intriguing systems in which the properties of thermo-responsive polymeric colloids are enriched by the presence of charged groups. In order to rationalize their properties and predict the behaviour of microgel suspensions, it is necessary to develop a coarse-graining strategy that starts from the accurate modelling of single particles. Here, we provide a numerical advancement of a recently-introduced model for charged co-polymerized microgels by improving the treatment of ionic groups in the polymer network. We investigate the thermoresponsive properties of the particles, in particular their swelling behaviour and structure, finding that, when charged groups are considered to be hydrophilic at all temperatures, highly charged microgels do not achieve a fully collapsed state, in favorable comparison to experiments. In addition, we explicitly include the solvent in the description and put forward a mapping between the solvophobic potential in the absence of the solvent and the monomer-solvent interactions in its presence, which is found to work very accurately for any charge fraction of the microgel. Our work paves the way for comparing single-particle properties and swelling behaviour of ionic microgels to experiments and to tackle the study of these charged soft particles at a liquid-liquid interface.

Modeling Microgels with a Controlled Structure across the Volume Phase Transition.

Thermoresponsive microgels are soft colloids that find widespread use as model systems for soft matter physics. Their complex internal architecture, made of a disordered and heterogeneous polymer network, has been so far a major challenge for computer simulations. In this work, we put forward a coarse-grained model of microgels whose structural properties are in quantitative agreement with results obtained with small-angle X-ray scattering experiments across a wide range of temperatures, encompassing the volume phase transition. These results bridge the gap between experiments and simulations of individual microgel particles, paving the way to theoretically address open questions about their bulk properties with unprecedented nano- and microscale resolution.

Numerical insights on ionic microgels: structure and swelling behaviour.

Recent progress has been made in the numerical modelling of neutral microgel particles with a realistic, disordered structure. In this work we extend this approach to the case of co-polymerised microgels where a thermoresponsive polymer is mixed with acidic groups. We compare the cases where counterions directly interact with microgel charges or are modelled implicitly through a Debye-Hückel description. We do so by performing extensive numerical simulations of single microgels across the volume phase transition (VPT) varying the temperature and the fraction of charged monomers. We find that the presence of charges considerably alters the microgel structure, quantified by the monomer density profiles and by the form factors of the microgels, particularly close to the VPT. We observe significant deviations between the implicit and explicit models, with the latter comparing more favourably to available experiments. In particular, we observe a shift of the VPT temperature to larger values as the amount of charged monomers increases. We also find that below the VPT the microgel-counterion complex is almost neutral, while it develops a net charge above the VPT. Interestingly, under these conditions the collapsed microgel still retains a large amount of counterions inside its structure. Since these interesting features cannot be captured by the implicit model, our results show that it is crucial to explicitly include the counterions in order to realistically model ionic thermoresponsive microgels.

Does the Adam-Gibbs relation hold in simulated supercooled liquids?

We perform stringent tests of thermodynamic theories of the glass transition over the experimentally relevant temperature regime for several simulated glass-formers. The swap Monte Carlo algorithm is used to estimate the configurational entropy and static point-to-set lengthscale, and careful extrapolations are used for the relaxation times. We first quantify the relation between configurational entropy and the point-to-set lengthscale in two and three dimensions. We then show that the Adam-Gibbs relation is generally violated in simulated models for the experimentally relevant time window. Collecting experimental data for several supercooled molecular liquids, we show that the same trends are observed experimentally. Deviations from the Adam-Gibbs relation remain compatible with random first order transition theory and may account for the reported discrepancies between Kauzmann and Vogel-Fulcher-Tammann temperatures. Alternatively, they may also indicate that even near Tg thermodynamics is not the only driving force for slow dynamics.

Zero-temperature glass transition in two dimensions.

Liquids cooled towards the glass transition temperature transform into amorphous solids that have a wide range of applications. While the nature of this transformation is understood rigorously in the mean-field limit of infinite spatial dimensions, the problem remains wide open in physical dimensions. Nontrivial finite-dimensional fluctuations are hard to control analytically, and experiments fail to provide conclusive evidence regarding the nature of the glass transition. Here, we develop Monte Carlo methods for two-dimensional glass-forming liquids that allow us to access equilibrium states at sufficiently low temperatures to directly probe the glass transition in a regime inaccessible to experiments. We find that the liquid state terminates at a thermodynamic glass transition which occurs at zero temperature and is associated with an entropy crisis and a diverging static correlation length. Our results thus demonstrate that a thermodynamic glass transition can occur in finite dimensional glass-formers.

Microgels Adsorbed at Liquid-Liquid Interfaces: A Joint Numerical and Experimental Study.

Soft particles display highly versatile properties with respect to hard colloids and even more so at fluid-fluid interfaces. In particular, microgels, consisting of a cross-linked polymer network, are able to deform and flatten upon adsorption at the interface due to the balance between surface tension and internal elasticity. Despite the existence of experimental results, a detailed theoretical understanding of this phenomenon is still lacking due to the absence of appropriate microscopic models. In this work, we propose an advanced modeling of microgels at a flat water/oil interface. The model builds on a realistic description of the internal polymeric architecture and single-particle properties of the microgel and is able to reproduce its experimentally observed shape at the interface. Complementing molecular dynamics simulations with in situ cryo-electron microscopy experiments and atomic force microscopy imaging after Langmuir-Blodgett deposition, we compare the morphology of the microgels for different values of the cross-linking ratios. Our model allows for a systematic microscopic investigation of soft particles at fluid interfaces, which is essential to develop predictive power for the use of microgels in a broad range of applications, including the stabilization of smart emulsions and the versatile patterning of surfaces.

Low-frequency vibrational modes of stable glasses.

Unusual features of the vibrational density of states D(ω) of glasses allow one to rationalize their peculiar low-temperature properties. Simulational studies of D(ω) have been restricted to studying poorly annealed glasses that may not be relevant to experiments. Here we report on D(ω) of zero-temperature glasses with kinetic stabilities ranging from poorly annealed to ultrastable glasses. For all preparations, the low-frequency part of D(ω) splits between extended and quasi-localized modes. Extended modes exhibit a boson peak crossing over to Debye behavior (Dex(ω) ~ ω2) at low-frequency, with a strong correlation between the two regimes. Quasi-localized modes obey Dloc(ω) ~ ω4, irrespective of the stability. The prefactor of this quartic law decreases with increasing stability, and the corresponding modes become more localized and sparser. Our work is the first numerical observation of quasi-localized modes in a regime relevant to experiments, and it establishes a direct connection between glasses' stability and their soft vibrational modes.

Configurational entropy measurements in extremely supercooled liquids that break the glass ceiling.

Liquids relax extremely slowly on approaching the glass state. One explanation is that an entropy crisis, because of the rarefaction of available states, makes it increasingly arduous to reach equilibrium in that regime. Validating this scenario is challenging, because experiments offer limited resolution, while numerical studies lag more than eight orders of magnitude behind experimentally relevant timescales. In this work, we not only close the colossal gap between experiments and simulations but manage to create in silico configurations that have no experimental analog yet. Deploying a range of computational tools, we obtain four estimates of their configurational entropy. These measurements consistently confirm that the steep entropy decrease observed in experiments is also found in simulations, even beyond the experimental glass transition. Our numerical results thus extend the observational window into the physics of glasses and reinforce the relevance of an entropy crisis for understanding their formation.

Equilibrium Sampling of Hard Spheres up to the Jamming Density and Beyond.

We implement and optimize a particle-swap Monte Carlo algorithm that allows us to thermalize a polydisperse system of hard spheres up to unprecedentedly large volume fractions, where previous algorithms and experiments fail to equilibrate. We show that no glass singularity intervenes before the jamming density, which we independently determine through two distinct nonequilibrium protocols. We demonstrate that equilibrium fluid and nonequilibrium jammed states can have the same density, showing that the jamming transition cannot be the end point of the fluid branch.

Observable-dependence of the effective temperature in off-equilibrium diatomic molecular liquids.

We discuss the observable-dependence of the effective temperature Teff, defined via the fluctuation-dissipation relation, of an out-of-equilibrium system composed by homonuclear dumbbell molecules. Teff is calculated by evaluating the fluctuation and the response for two observables associated, respectively, to translational and to rotational degrees of freedom, following a sudden temperature quench. We repeat our calculations for different dumbbell elongations ζ. At high elongations (ζ > 0.4), we find the same Teff for the two observables. At low elongations (ζ ⩽ 0.4), only for very deep quenches Teff coincides. The observable-dependence of Teff for low elongations and shallow quenches stresses the importance of a strong coupling between orientational and translational variables for a consistent definition of the effective temperature in glassy systems.