Connecting Elasticity and Effective Interactions of Neutral Microgels: The Validity of the Hertzian Model

Core-shell nanogels: the effects of morphology, electro- and magnetostatic interactions

We study the influence of core-shell morphology on the structural characteristics of nanogels. Using computer simulations, we examine three different types of systems, distinguished by their intermonomer interactions: those with excluded volume only; those with charged monomers and excluded volume; and those with excluded volume combined with a certain number of magnetised nanoparticles incorporated within the nanogel. We observe that if the polymers in the shell are short and dense, they tend to penetrate the core. This effect of backfolding is enhanced in charged nanogels, regardless of whether all monomers are charged, or only the core or shell ones. The presence of an experimentally available amount of magnetic nanoparticles in a gel, on the one hand, does not lead to any significant morphological changes. On the other hand, the morphology of the nanogel with magnetic particles has an impact on its magnetic susceptibility. Particular growth of the magnetic response is observed if a long shell of a nanogel is functionalised.

Compressive elasticity of epoxy functionalized Chitosan-based semi-IPN cryobeads of N-alkyl methacrylate esters: Validity of the Hertzian model with experiments

In situ forming poly(dimethylaminoethyl methacrylate-co-glycidylmethacrylate)/Chitosan, P(DMAEMA-co-GMA)/Chitosan, (PDG/CS) cryobeads based on "dropwise freezing into cryogenic liquid method" combined with "blending with polymer method" are promising for applications due to their pH-responsiveness and stability under physiological conditions. Based on classical contact mechanics, Hertzian elasticity of semi-interpenetrated network (semi-IPN) cryobeads was analyzed to examine whether there is a direct correlation between elastic properties of single particle and its macroscopic behavior. A one-step procedure has been proposed to design chitosan-interpenetrated cryobeads with a cationic nature via combination of structural properties as well as functionality of chitosan containing primary and secondary hydroxyl and amino groups. The study is focused on characterization of network formation kinetics in different shapes and how different production variables affect the elasticity/swelling performance of cross-linked system. The elastic properties of semi-IPN cryobeads were improved by both adding chitosan to copolymer PDG structure and lowering the gelation temperature to cryogelation conditions. The results obtained highlighted the importance of composition to modulate elasticity, the influence of preparation temperature and shape of cryobeads on their elasticity. Findings regarding the topography-dependent local elastic properties of chitosan-incorporated semi-IPN gels offer possibilities for modulating the behavior of chitosan-based soft materials.

Real and In Silico Microgels Show Comparable Bulk Moduli Below and Above the Volume Phase Transition

The compressibility of soft colloids influences their phase behavior and flow properties, especially in concentrated suspensions. Particle compressibility, which is proportional to the reciprocal of the bulk modulus K, is a key parameter for soft polymer-based particles that can be compressed in crowded environments. Here, microgels with different degrees of cross-linking, i.e., softness, are investigated below and above their volume phase transition temperature (VPTT). By combining molecular dynamics simulations with small-angle neutron scattering with contrast variation, a change in the particle bulk moduli of two orders of magnitude is observed. The degree of cross-linking has a significant impact on the bulk modulus of the swollen microgel, while above the VPTT the values of K are almost independent of the cross-linking density. The excellent agreement between experimental results and simulations also highlight that the model microgels from computer simulations possess both the internal architecture and the elastic properties of real polymeric networks. This paves the way to a systematic use of simulations to investigate the behavior of dense microgel suspensions below and above their VPTT.

External field-driven property localization in liquids of responsive macromolecules

We explore theoretically the effects of external potentials on the spatial distribution of particle properties in a liquid of explicitly responsive macromolecules. In particular, we focus on the bistable particle size as a coarse-grained internal degree of freedom (DoF, or "property"), σ, that moves in a bimodal energy landscape, in order to model the response of a state-switching (big-to-small) macromolecular liquid to external stimuli. We employ a mean-field density functional theory (DFT) that provides the full inhomogeneous equilibrium distributions of a one-component model system of responsive colloids (RCs) interacting with a Gaussian pair potential. For systems confined between two parallel hard walls, we observe and rationalize a significant localization of the big particle state close to the walls, with pressures described by an exact RC wall theorem. Application of more complex external potentials, such as linear (gravitational), osmotic, and Hamaker potentials, promotes even stronger particle size segregation, in which macromolecules of different size are localized in different spatial regions. Importantly, we demonstrate how the degree of responsiveness of the particle size and its coupling to the external potential tune the position-dependent size distribution. The DFT predictions are corroborated by Brownian dynamics simulations. Our study highlights the fact that particle responsiveness can be used to localize liquid properties and therefore helps to control the property- and position-dependent function of macromolecules, e.g., in biomedical applications.

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.

Concentration and temperature dependent interactions and state diagram of dispersions of copolymer microgels

We investigate by means of small angle neutron scattering experiments and numerical simulations the interactions and inter-particle arrangements of concentrated dispersions of copolymer poly(N-isopropylacrylamide)-poly(ethylene glycol methyl ether methacrylate) (PNIPAM-PEGMA) microgels across the volume phase transition (VPT). The scattering data of moderately concentrated dispersions are accurately modeled at all temperatures by using a star polymer form factor and static structure factors calculated from the effective potential obtained from simulations. Interestingly, for temperatures below the VPT temperature (VPTT), the radius of gyration and blob size of the particles significantly decrease with increasing the effective packing fraction in the non-overlapping regime. This is attributed to the presence of charges in the system associated with the use of an ionic initiator in the synthesis. Simulations using the experimentally corroborated interaction potential are used to explore the state diagram in a wide range of effective packing fractions. Below and slightly above the VPTT, the system undergoes an arrest transition mainly driven by the soft repulsion between the particles. Only well above the VPTT the system is found to phase separate before arresting. Our results highlight the versatility and potential of copolymer PNIPAM-PEGMA microgels to explore different kinds of arrested states balancing attraction and repulsion by changing temperature and packing fraction.

Buckling and Interfacial Deformation of Fluorescent Poly(N-isopropylacrylamide) Microgel Capsules

Hollow microgels are fascinating model systems at the crossover between polymer vesicles, emulsions, and colloids as they deform, interpenetrate, and eventually shrink at higher volume fraction or when subjected to an external stress. Here, we introduce a system consisting of microgels with a micrometer-sized cavity enabling a straightforward characterization in situ using fluorescence microscopy techniques. Similarly to elastic capsules, these systems are found to reversibly buckle above a critical osmotic pressure, conversely to smaller hollow microgels, which were previously reported to deswell at high volume fraction. Simulations performed on monomer-resolved in silico hollow microgels confirm the buckling transition and show that the presented microgels can be described with a thin shell model theory. When brought to an interface, these microgels, that we define as microgel capsules, strongly deform and we thus propose to utilize them to locally probe interfacial properties within a theoretical framework adapted from the Johnson-Kendall-Roberts (JKR) theory. Besides their capability to sense their environment and to address fundamental questions on the elasticity and permeability of microgel systems, microgel capsules can be further envisioned as model systems mimicking anisotropic responsive biological systems such as red blood and epithelial cells thanks to the possibility offered by microgels to be synthesized with custom-designed properties.

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.

Building block properties govern granular hydrogel mechanics through contact deformations

Granular hydrogels have been increasingly exploited in biomedical applications, including wound healing and cardiac repair. Despite their utility, design guidelines for engineering their macroscale properties remain limited, as we do not understand how the properties of granular hydrogels emerge from collective interactions of their microgel building blocks. In this work, we related building block features (stiffness and size) to the macroscale properties of granular hydrogels using contact mechanics. We investigated the mechanics of the microgel packings through dynamic oscillatory rheology. In addition, we modeled the system as a collection of two-body interactions and applied the Zwanzig and Mountain formula to calculate the plateau modulus and viscosity of the granular hydrogels. The calculations agreed with the dynamic mechanical measurements and described how microgel properties and contact deformations define the rheology of granular hydrogels. These results support a rational design framework for improved engineering of this fascinating class of materials.

Beyond simple self-healing: How anisotropic nanogels adapt their shape to their environment

The response of soft colloids to crowding depends sensitively on the particles’ compressibility. Nanogel suspensions provide model systems that are often studied to better understand the properties of soft materials and complex fluids from the formation of colloidal crystals to the flow of viruses, blood, or platelet cells in the body. Large spherical nanogels, when embedded in a matrix of smaller nanogels, have the unique ability to spontaneously deswell to match their size to that of the nanogel composing the matrix. In contrast to hard colloids, this self-healing mechanism allows for crystal formation without giving rise to point defects or dislocations. Here, we show that anisotropic ellipsoidal nanogels adapt both their size and their shape depending on the nature of the particles composing the matrix in which they are embedded. Using small-angle neutron scattering with contrast variation, we show that ellipsoidal nanogels become spherical when embedded in a matrix of spherical nanogels. In contrast, the anisotropy of the ellipsoid is enhanced when they are embedded in a matrix of anisotropic nanogels. Our experimental data are supported by Monte Carlo simulations that reproduce the trend of decreasing aspect ratio of ellipsoidal nanogels with increasing crowding by a matrix of spherical nanogels.

Active responsive colloids coupled to different thermostats

We introduce a model of active responsive colloids (ARCs) in which an internal degree of freedom (DoF) of a single colloidal particle is "activated" by coupling it to a different thermostat than for the translational DoFs. As for the responsive internal DoF, we consider specifically the size (diameter) of the spherical particles, which is confined by a harmonic parent potential being either entropic or energetic in nature. The ARCs interact via a repulsive Hertzian pair potential, appropriate to model hydrogels or elastic colloids, and are studied for various densities using Brownian dynamics simulations. We tune the internal activity in the nonequilibrium steady state by scanning through a wide range of internal temperatures, both smaller ("colder") and larger ("hotter") than the translational temperature. The results show a rich and intriguing behavior for the emergent property distributions, colloidal pair structure, and the diffusive translational dynamics controlled by the internal activity, substantially depending on whether the internal DoF moves in an entropic or energetic potential. We discuss theoretical thermal limits and phenomenological models which can explain some of the nonequilibrium trends qualitatively. Our study indicates that particle dynamical polydispersity as well as the structure and dynamics of dense macromolecular suspensions can be vastly tuned by internal activity in terms of internal "hot" or "cold" fluctuating states.

Resolving the different bulk moduli within individual soft nanogels using small-angle neutron scattering

The bulk modulus, K, quantifies the elastic response of an object to an isotropic compression. For soft compressible colloids, knowing K is essential to accurately predict the suspension response to crowding. Most colloids have complex architectures characterized by different softness, which additionally depends on compression. Here, we determine the different values of K for the various morphological parts of individual nanogels and probe the changes of K with compression. Our method uses a partially deuterated polymer, which exerts the required isotropic stress, and small-angle neutron scattering with contrast matching to determine the form factor of the particles without any scattering contribution from the polymer. We show a clear difference in softness, compressibility, and evolution of K between the shell of the nanogel and the rest of the particle, depending on the amount of cross-linker used in their synthesis.

How Softness Matters in Soft Nanogels and Nanogel Assemblies

Softness plays a key role in determining the macroscopic properties of colloidal systems, from synthetic nanogels to biological macromolecules, from viruses to star polymers. However, we are missing a way to quantify what the term "softness" means in nanoscience. Having quantitative parameters is fundamental to compare different systems and understand what the consequences of softness on the macroscopic properties are. Here, we propose different quantities that can be measured using scattering methods and microscopy experiments. On the basis of these quantities, we review the recent literature on micro- and nanogels, i.e. cross-linked polymer networks swollen in water, a widely used model system for soft colloids. Applying our criteria, we address the question what makes a nanomaterial soft? We discuss and introduce general criteria to quantify the different definitions of softness for an individual compressible colloid. This is done in terms of the energetic cost associated with the deformation and the capability of the colloid to isotropically deswell. Then, concentrated solutions of soft colloids are considered. New definitions of softness and new parameters, which depend on the particle-to-particle interactions, are introduced in terms of faceting and interpenetration. The influence of the different synthetic routes on the softness of nanogels is discussed. Concentrated solutions of nanogels are considered and we review the recent results in the literature concerning the phase behavior and flow properties of nanogels both in three and two dimensions, in the light of the different parameters we defined. The aim of this review is to look at the results on micro- and nanogels in a more quantitative way that allow us to explain the reported properties in terms of differences in colloidal softness. Furthermore, this review can give researchers dealing with soft colloids quantitative methods to define unambiguously which softness matters in their compound.

Changes in the Form Factor and Size Distribution of Nanogels in Crowded Environments

Particle size disparities suppress crystallization. However, soft deformable nanogels can change the size of the larger particles in suspension and crystallize even at a high initial size-polydispersity. Using neutron scattering with contrast variation, the response of individual nanogels in crowded environments was probed, and an increase of the parameter describing size-polydispersity was found, which is often interpreted as deformation. Here, computer simulations are used to generate deformed nanogels and the corresponding form factor. The data are fitted with the spherical model used to analyze scattering data. The fits show the same qualitative increase of the parameter related to the size-polydispersity with increasing particle deformation. Starting from the simulated deformed spheres, we also reproduce experimental scattering data. A further analysis of the particle shows that the size disparities between nanogels do not increase significantly. In contrast, their shapes strongly vary from one nanogel to the other.

Microgels react to force: mechanical properties, syntheses, and force-activated functions

Microgels are colloidal polymer networks with high molar mass and properties between rigid particles, flexible macromolecules, and micellar aggregates. Their unique stimuli-responsiveness in conjunction with their colloidal phase behavior render them useful for many applications ranging from engineering to biomedicine. In many scenarios either the microgel's mechanical properties or its interactions with mechanical force play an important role. Here, we firstly explain microgel mechanical properties and how these are measured by atomic force microscopy (AFM), then we equip the reader with the synthetic background to understand how specific architectures and chemical functionalities enable these mechanical properties, and eventually we elucidate how the interaction of force with microgels can lead to the activation of latent functionality. Since the interaction of microgels with force is a multiscale and multidisciplinary subject, we introduce and interconnect the different research areas that contribute to the understanding of this emerging field in this Tutorial Review.

Modulating internal transition kinetics of responsive macromolecules by collective crowding

Packing and crowding are used in biology as mechanisms to (self-)regulate internal molecular or cellular processes based on collective signaling. Here, we study how the transition kinetics of an internal "switch" of responsive macromolecules is modified collectively by their spatial packing. We employ Brownian dynamics simulations of a model of Responsive Colloids, in which an explicit internal degree of freedom-here, the particle size-moving in a bimodal energy landscape self-consistently responds to the density fluctuations of the crowded environment. We demonstrate that populations and transition times for the two-state switching kinetics can be tuned over one order of magnitude by "self-crowding." An exponential scaling law derived from a combination of Kramers' and liquid state perturbation theory is in very good agreement with the simulations.

On the role of softness in ionic microgel interactions

Thermoresponsive microgels are a popular model system to study phase transitions in soft matter, because temperature directly controls their volume fraction. Ionic microgels are additionally pH-responsive and possess a rich phase diagram. Although effective interaction potentials between microgel particles have been proposed, these have never been fully tested, leading to a gap in our understanding of the link between single-particle and collective properties. To help resolve this gap, four sets of ionic microgels with varying crosslinker density were synthesised and characterised using light scattering techniques and confocal microscopy. The resultant structural and dynamical information was used to investigate how particle softness affects the phase behaviour of ionic microgels and to validate the proposed interaction potential. We find that the architecture of the microgel plays a marked role in its phase behaviour. Rather than the ionic charges, it is the dangling ends which drive phase transitions and interactions at low concentration. Comparison to theory underlines the need for a refined theoretical model which takes into consideration these close-contact interactions.

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

Microgels, colloidal-scale polymer networks, are the prototype soft colloids. When the constituent polymers are thermoresponsive, they undergo a volume phase transition (VPT) from a swollen to a collapsed state at a characteristic temperature, close to ambient one, of great appeal for several applications. To describe this phenomenon, microgels are usually treated as neutral, but here we show that electrostatics needs to be taken into account. In particular, deswelling occurs via a two-step, rather than a homogeneous, particle collapse, mainly driven by peripheral charges located on the microgel corona, for which we also establish a unifying framework encompassing all studied microgels. Our work thus provides a change of perspective to describe these fascinating systems.

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.

Softness mapping of the concentration dependence of the dynamics in model soft colloidal systems

The dynamics of a series of soft colloids comprised of polystyrene cores with poly(N-isopropylacrylamide) (PNIPAM) coronas was investigated by diffusing wave spectroscopy (DWS). The modulus of the coronas was varied by changing the cross-link density and we were able to interpret the results within a hard-soft mapping framework. The soft, swellable particle properties were modeled using an extended Flory-Rehner theory and a Hertzian pair potential. Following volume fraction jumps, softness effects on the concentration dependence of dynamics were determined, with a 'soft colloids make strong glass-forming liquid'-type of behavior observed close to the nominal glass transition volume fraction, φ_g. Such behavior from the current systems cannot be fully explained by the osmotic deswelling model alone. However, inspired by the soft-hard mapping from Schmiedeberg et al, [Europhys. Lett. 2011, 96(3), 36010] we estimated effective hard-sphere diameters and achieved a successful mapping of the α-relaxation times to a master curve below φ_g. Above φ_g, the curves no longer collapse but show strong deviations from a Vogel-Fulcher type of divergence onto soft jamming plateaux. Our results provide evidence that osmotic deswelling itself cannot fully explain the observed dynamics. Softness also plays an important role in the dynamics of soft, concentrated colloids.

Defects in crystals of soft colloidal particles

In this paper we use computer simulations to examine point defects in systems of "soft" colloidal particles including Hertzian spheres, and star polymers. We use Monte Carlo simulations to determine the deformation of the different crystals associated with vacancies and interstitials and use thermodynamic integration to predict the equilibrium concentrations of such defects. We find that the nature of the lattice distortion is mainly determined by the crystal structure and not by the specifics of the interaction potential. We can distinguish one-, two-, and three-dimensional lattice distortions and find that the range of the distortion generally depends on the dimensionality. We find that in both model systems the deformation of the body-centered cubic (BCC) crystal caused by an interstitial is one-dimensional and we show that its structure is well described as a crowdion. Similarly, we show that the one-dimensional deformation of the hexagonal (H) crystal of Hertzian spheres caused by a vacancy can be characterized as a voidion. Interestingly, with the exception of the FCC crystal in the Hertzian sphere model, in all cases we find that the interstitial concentration is higher than the vacancy concentration. Most noteworthy, the concentration of interstitials in the BCC crystals can reach up to 1%.

Interface-induced hysteretic volume phase transition of microgels: simulation and experiment

Thermo-responsive microgel particles can exhibit a drastic volume shrinkage upon increasing the solvent temperature. Recently we found that the spreading of poly(N-isopropylacrylamide) (PNiPAm) microgels at a liquid interface under the influence of surface tension hinders the temperature-induced volume phase transition. In addition, we observed a hysteresis behavior upon temperature cycling, i.e. a different evolution in microgel size and shape depending on whether the microgel was initially adsorbed to the interface in expanded or collapsed state. Here, we model the volume phase transition of such microgels at an air/water interface by monomer-resolved Brownian dynamics simulations and compare the observed behavior with experiments. We reproduce the experimentally observed hysteresis in the microgel dimensions upon temperature variation. Our simulations did not observe any hysteresis for microgels dispersed in the bulk liquid, suggesting that it results from the distinct interfacial morphology of the microgel adsorbed at the liquid interface. An initially collapsed microgel brought to the interface and subjected to subsequent swelling and collapsing (resp. cooling and heating) will end up in a larger size than it had in the original collapsed state. Further temperature cycling, however, only shows a much reduced hysteresis, in agreement with our experimental observations. We attribute the hysteretic behavior to a kinetically trapped initial collapsed configuration, which relaxes upon expanding in the swollen state. We find a similar behavior for linear PNiPAm chains adsorbed to an interface. Our combined experimental - simulation investigation provides new insights into the volume phase transition of PNiPAm materials adsorbed to liquid interfaces.

Characterization of the volume fraction of soft deformable microgels by means of small-angle neutron scattering with contrast variation

The volume occupied by colloids in a suspension - namely the volume fraction - is the thermodynamic variable that determines the phase behavior of these systems. While for hard incompressible spheres this quantity is well defined, for soft compressible colloids such as microgels - polymeric crosslinked networks swollen in a good solvent - the determination of the real volume occupied by these particles in solution is particularly challenging. This fact depends on two aspects: first the surface and, therefore, the volume of the microgels is hard to define properly given their external fuzziness; second, microgels can osmotically deswell, deform or interpenetrate their neighbors, i.e. change their shape and size depending on the solution concentration. Here, the form factors of few hydrogenated microgels embedded in a matrix of deuterated but otherwise identical microgels are measured using small-angle neutron scattering with contrast variation. From the analysis of the scattering data, the variation of the volume of the microgels as a function of concentration is obtained and used to calculate the real microgel volume fraction in solution. Soft neutral microgels are shown to facet already at low concentrations while in contrast, harder microgels maintain their shape and change their volume.

Linear and nonlinear viscoelasticity of concentrated thermoresponsive microgel suspensions

We present an integrated experimental and theoretical study of the dynamics and rheology of self-crosslinked, slightly charged, temperature responsive soft poly(N-isopropylacrylamide) (pNIPAM) microgels over a wide range of concentration and temperature spanning the sharp change in particle size and intermolecular interactions across the lower critical solution temperature (LCST). Dramatic, non-monotonic changes in viscoelasticity are observed as a function of temperature, with distinct concentration dependence in the dense fluid, glassy, and soft-jammed regimes. Motivated by our experimental observations, we formulate a minimalistic model for the size dependence of a single microgel particle and the change of the interparticle interaction from purely repulsive to attractive upon heating. Using microscopic equilibrium and time-dependent statistical mechanical theories, theoretical predictions are quantitatively compared with experimental measurements of the shear modulus. Good agreement is found for the nonmonotonic temperature behavior that originates as a consequence of the competition between reduced microgel packing fraction and increasing interparticle attractions. Testable predictions are made for nonlinear rheological properties such as the yield stress and strain. To our knowledge, this is the first attempt to quantitatively understand in a unified manner the viscoelasticity of dense, temperature-responsive microgel suspensions spanning a wide range of temperatures and concentrations.

Viscoelasticity of non-colloidal hydrogel particle suspensions at the liquid-solid transition

Suspensions of soft particles transition from a viscous fluid to a soft material upon increases in phase volume. The criteria defining the transition to this jammed state are difficult to define due to the porous and deformable nature of soft particles. Here, we characterise the rheology of aqueous suspensions of industrially relevant non-colloidal, polydisperse, frictional agarose microgels and evaluate shear and viscoelastic behaviour across a range of phase volumes from the dilute regime to the highly concentrated regime. In order to model the viscoelastic response of suspensions without free fitting parameters, the random close packing volume fraction (φrcp) and the particle modulus are determined, respectively, from particle size distribution measurements and direct measurements of reduced elastic modulus of individual particles (Erp) using Atomic Force Microscopy. It is found that at φrcp, previously shown to correspond to divergence of the viscosity, also corresponds to the suspension transition from a viscous to viscoelastic fluid. However, the transition to a jammed solid-like state (φj) occurs at phase volumes exceeding this value (i.e. φj > φrcp). The suspension modulus and its sudden growth at φj are well-predicted by the Evans and Lips model that incorporates the Erp of the hydrogel particles. This rheological behaviour showing a dual transition is reminiscent of two families of systems: (i) colloidal suspensions and (ii) frictional-adhesive non-colloidal suspensions. However, it does not strictly follow either case. We propose that the width of the transition region is dictated by frictional contact, particle size distribution and particle modulus, and plan to further probe this in future work.

Soft Particles at Liquid Interfaces: From Molecular Particle Architecture to Collective Phase Behavior

Soft particles such as microgels can undergo significant and anisotropic deformations when adsorbed to a liquid interface. This, in turn, leads to a complex phase behavior upon compression. To date, experimental efforts have predominantly provided phenomenological links between microgel structure and resulting interfacial behavior, while simulations have not been entirely successful in reproducing experiments or predicting the minimal requirements for the desired phase behavior. Here, we develop a multiscale framework to link the molecular particle architecture to the resulting interfacial morphology and, ultimately, to the collective interfacial phase behavior. To this end, we investigate interfacial morphologies of different poly(N-isopropylacrylamide) particle systems using phase-contrast atomic force microscopy and correlate the distinct interfacial morphology with their bulk molecular architecture. We subsequently introduce a new coarse-grained simulation method that uses augmented potentials to translate this interfacial morphology into the resulting phase behavior upon compression. The main novelty of this method is the possibility to efficiently encode multibody interactions, the effects of which are key to distinguishing between heterostructural (anisotropic collapse) and isostructural (isotropic collapse) phase transitions. Our approach allows us to qualitatively resolve existing discrepancies between experiments and simulations. Notably, we demonstrate the first in silico account of the two-dimensional isostructural transition, which is frequently found in experiments but elusive in simulations. In addition, we provide the first experimental demonstration of a heterostructural transition to a chain phase in a single-component system, which has been theoretically predicted decades ago. Overall, our multiscale framework provides a phenomenological bridge between physicochemical soft-particle characteristics at the molecular scale and nanoscale and the collective self-assembly phenomenology at the macroscale, serving as a stepping stone toward an ultimately more quantitative and predictive design approach.

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.

Structure and dynamics of responsive colloids with dynamical polydispersity

Dynamical polydispersity in single-particle properties, for example a fluctuating particle size, shape, charge density, etc, is intrinsic to responsive colloids (RCs), such as biomacromolecules or microgels, but is typically not resolved in coarse-grained mesoscale simulations. Here, we present Brownian dynamics simulations of suspensions of RCs modeling soft hydrogel colloids, for which the size of the individual particles is an explicitly resolved (Gaussian) degree of freedom and dynamically responds to the local interacting environment. We calculate the liquid structure, emergent size distributions, long-time diffusion, and property (size) relaxation kinetics for a wide range of densities and intrinsic property relaxation times in the canonical ensemble. Comparison to interesting reference cases, such as conventional polydisperse suspensions with a frozen parent distribution, or conventional monodisperse systems interacting with an effective pair potential for one fixed size, shows a significant spread in the structure and dynamics. The differences, most apparent in the high density regimes, are due to many-body correlations and the dynamical coupling between property and translation in RC systems, not explicitly accounted for in the conventional treatments. In particular, the translational diffusion in the RC systems is surprisingly close to the free (single RC) diffusion, mainly due to a cancellation of crowding and size compression effects. We show that an effective monodisperse pair potential can be constructed that describes the many-body correlations reasonably well by convoluting the RC pair potential with the density-dependent emergent size distributions and using a mean effective diffusion constant.

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.

Osmotic pressure of suspensions comprised of charged microgels

We determine the osmotic pressure of microgel suspensions using membrane osmometry and dialysis, for microgels with different softnesses. Our measurements reveal that the osmotic pressure of solutions of both ionic and neutral microgels is determined by the free ions that leave the microgel periphery to maximize their entropy and not by the translational degrees of freedom of the microgels themselves. Furthermore, up to a given concentration it is energetically favorable for the microgels to maintain a constant volume without appreciable deswelling. The concentration where deswelling starts weakly depends on the crosslinker concentration, which affects the microgel dimension; we explain this by considering the dependence of the osmotic pressure and the microgel bulk modulus on the particle size.

Hyperbranched DNA clusters

Taking advantage of the base-pairing specificity and tunability of DNA interactions, we investigate the spontaneous formation of hyperbranched clusters starting from purposely designed DNA tetravalent nanostar monomers, encoding in their four sticky ends the desired binding rules. Specifically, we combine molecular dynamics simulations and Dynamic Light Scattering experiments to follow the aggregation process of DNA nanostars at different concentrations and temperatures. At odds with the Flory-Stockmayer predictions, we find that, even when all possible bonds are formed, the system does not reach percolation due to the presence of intracluster bonds. We present an extension of the Flory-Stockmayer theory that properly describes the numerical and experimental results.

Structure and position-dependent properties of inhomogeneous suspensions of responsive colloids

Responsive particles, such as biomacromolecules or hydrogels, display a broad and polymodal distribution of conformations and have thus the ability to change their properties (e.g., size, shape, charge density, etc.) substantially in response to external fields or to their local environment (e.g., mediated by cosolutes or pH). Here we discuss the basic statistical mechanics for a model of responsive colloids (RCs) by introducing an additional "property" degree of freedom as a collective variable in a formal coarse-graining procedure. The latter leads to an additional one-body term in the coarse-grained (CG) free energy, defining a single-particle property distribution for an individual polydisperse RC. We argue that in the equilibrium thermodynamic limit such a CG system of RCs behaves like a conventional polydisperse system of nonresponsive particles. We then illustrate the action of external fields, which impose local (position-dependent) property distributions leading to nontrivial effects on the spatial one-body property and density profiles, even for an ideal (noninteracting) gas of RCs. We finally apply density-functional theory in the local density approximation to discuss the effects of particle interactions for specific examples of (i) a suspension of RCs in an external field linear in both position and property, (ii) a suspension of RCs with highly localized properties (sizes) confined between two walls, and (iii) a two-component suspension where an inhomogeneously distributed (nonresponsive) cosolute component, as found, e.g., in the studies of osmolyte- or salt-induced collapse or swelling transitions of thermosensitive polymers, modifies the local properties and density of the RC liquid.

Evidence of Structural Inhomogeneities in Hard-Soft Dimeric Particles without Attractive Interactions

We perform Monte Carlo simulations of a simple hard-soft dimeric model constituted by two tangent spheres experiencing different interactions. Specifically, two hard spheres belonging to different dimers interact via a bare hard-core repulsion, whereas two soft spheres experience a softly repulsive Hertzian interaction. The cross correlations are soft as well. By exploring a wide range of temperatures and densities we investigate the capability of this model to document the existence of structural inhomogeneities indicating the possible onset of aggregates, even if no attraction is set. The fluid phase behavior is studied by analyzing structural and thermodynamical properties of the observed structures, in particular by computing radial distribution functions, structure factors and cluster size distributions. The numerical results are supported by integral equation theories of molecular liquids which allow for a finer and faster spanning of the temperature-density diagram. Our results may serve as a framework for a more systematic investigation of self-assembled structures of functionalized hard-soft dimers able to aggregate in a variety of structures widely oberved in colloidal dispersion.

Monte Carlo simulation and integral equation study of Hertzian spheres in the low-temperature regime

We investigate the behavior of Hertzian spheres in the fluid phase and in proximity of the freezing threshold by using Monte Carlo (MC) simulations and integral equation theories, based on the Ornstein-Zernike (OZ) approach. The study is motivated by the importance of the Hertzian model in representing a large class of systems interacting via soft interactions, such as star polymers or microgels. Radial distribution functions, structure factors, and excess entropy clearly show the reentrant behavior typical of the Hertzian fluid, well caught by both MC simulations and OZ theory. Then, we make use of some phenomenological one-phase criteria for testing their reliability in detecting the freezing threshold. All criteria provide evidence of the fluid-solid transition with different degrees of accuracy. This suggests the possibility to adopt these empirical rules to provide a quick and reasonable estimate of the freezing transition in model potentials of wide interest for soft matter systems.

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.

Multi-particle collision dynamics for a coarse-grained model of soft colloids

The growing interest in the dynamical properties of colloidal suspensions, both in equilibrium and under an external drive such as shear or pressure flow, requires the development of accurate methods to correctly include hydrodynamic effects due to the suspension in a solvent. In the present work, we generalize Multiparticle Collision Dynamics (MPCD) to be able to deal with soft, polymeric colloids. Our methods build on the knowledge of the monomer density profile that can be obtained from monomer-resolved simulations without hydrodynamics or from theoretical arguments. We hereby propose two different approaches. The first one simply extends the MPCD method by including in the simulations effective monomers with a given density profile, thus neglecting monomer-monomer interactions. The second one considers the macromolecule as a single penetrable soft colloid (PSC), which is permeated by an inhomogeneous distribution of solvent particles. By defining an appropriate set of rules to control the collision events between the solvent and the soft colloid, both linear and angular momenta are exchanged. We apply these methods to the case of linear chains and star polymers for varying monomer lengths and arm number, respectively, and compare the results for the dynamical properties with those obtained within monomer-resolved simulations. We find that the effective monomer method works well for linear chains, while the PSC method provides very good results for stars. These methods pave the way to extend MPCD treatments to complex macromolecular objects such as microgels or dendrimers and to work with soft colloids at finite concentrations.