Microgels at Interfaces Behave as 2D Elastic Particles Featuring Reentrant Dynamics

Soft colloids are increasingly used as model systems to address fundamental issues such as crystallization and the glass and jamming transitions. Among the available classes of soft colloids, microgels are emerging as the gold standard. Since their great internal complexity makes their theoretical characterization very hard, microgels are commonly modeled, at least in the small-deformation regime, within the simple framework of linear elasticity theory. Here we show that there exist conditions where its range of validity can be greatly extended, providing strong numerical evidence that microgels adsorbed at an interface follow the two-dimensional Hertzian theory, and hence behave like 2D elastic particles, up to very large deformations, in stark contrast to what found in bulk conditions. We are also able to estimate Young’s modulus of the individual particles and, by comparing it with its counterpart in bulk conditions, we demonstrate a significant stiffening of the polymer network at the interface. Finally, by analyzing dynamical properties, we predict multiple reentrant phenomena: By a continuous increase of particle density, microgels first arrest and then refluidify due to the high penetrability of their extended coronas. We observe this anomalous behavior in a range of experimentally accessible conditions for small and loosely cross-linked microgels. The present work thus establishes microgels at interfaces as a new model system for fundamental investigations, paving the way for the experimental synthesis and research on unique high-density liquidlike states. In addition, these results can guide the development of novel assembly and patterning strategies on surfaces and the design of novel materials with desired interfacial behavior.

Popular Summary

The properties and the structure of colloids—in which particles of one substance are dispersed in another—are determined by the way those particles interact with each other. An easy guess might lead one to say that complex particles possess an equally complex interaction potential. However, this may not always be the case. Here, we consider microgels, which are colloids made of an intricate polymeric architecture, confined at a liquid-liquid interface. We show that this complex situation can be described in terms of a collection of simple 2D elastic objects.

This simplicity is not necessarily linked to the absence of intriguing behavior but actually reveals the presence of multiple reentrant transitions of the system at high densities. This means that, unlike most dense liquids, diffusion speeds up when the particle concentration increases. Remarkably, we demonstrate that this peculiar behavior is within accessible experimental conditions of small and soft microgels. In addition, the presence of the interface itself has consequences for the stiffness of the polymer network. Through our numerical study, we find that the polymer chains of the microgel are more stretched and less responsive than in bulk.

Altogether, our work suggests the use of microgels on the collective scale as an ideal system for fundamental investigations in two dimensions with unique reentrant dynamics. Besides, the a priori knowledge of the interactions that we provide here will allow for a clever design of microgel assemblies with desired properties.

Figure 1

Microgels interacting at the interface. Top and side simulation snapshots of two microgels with $c=5\%$ at the water-oil interface at a representative distance $r\approx 40{\sigma }_m$. The effective diameter of the microgel is ${\sigma }_{\mathrm{eff}}$. Solvent particles are not shown for clarity.

Figure 2

Effective potentials for microgels at the liquid-liquid interface and related 2D Hertzian fit parameters. (a) Numerical results refer to three values of the cross-linker concentrations $c$: 3% (circles), 5% (squares), and 10% (triangles). Full lines are fits to numerical results using Eq. (1). The two vertical dotted lines indicate the value ${\sigma }_{\mathrm{eff}}\approx 53{\sigma }_m$ and the distance at which the cores of the two microgels get in contact ($r\approx 30{\sigma }_m$) for $c=5\%$; (b) Microgel effective diameter ${\sigma }_{\mathrm{eff}}$ compared to the extension on the plane of the interface ${\sigma }_{\mathrm{ext}}$ calculated as in Ref. [29], in units of ${\sigma }_m$; (c) Young’s modulus $Y$ extracted from the 2D Hertzian fit to $V_H$ in units of $k_{B}T/{\sigma }_m^2$.

Figure 3

Elastic moduli at the interface and in the bulk. Bulk modulus $K$, shear modulus $G$, Poisson’s ratio $\nu$, and Young’s modulus $Y$ for the same microgel topology at the interface (full symbols) and in bulk (empty symbols) with explicit solvent as a function of $c$. In the last row, the theoretical results for $Y$ are also compared to the ones obtained from the effective potential fits with the Hertzian model [Eq. (1)], also reported in Fig. 2. $K$, $G$, and $Y$ are in units of $k_{B}T/{\sigma }_m^3$ to appropriately compare bulk and interface moduli, where the latter moduli are divided by the thickness of the shell at the interface (approximately ${\sigma }_m$); $\nu$ is dimensionless. Error bars estimated from the fits of $P(J)$ for $K$ and $P(I)$ for $G$ (see Sec. 5) are propagated in the calculation of $\nu$ and $Y$.

Figure 4

2D Hertzian phase diagram. (a) Diffusion coefficient $D$ as a function of the area fraction $\varphi$ for different values of the 2D Hertzian strength $A$. From top to bottom, $A$ takes the following values: 226, 340, 409, 453, 566, 680, 793, 906, 974, 1042, $1133k_{B}T$; symbols are simulation data and lines serve as guides to the eye. The lowest reported value of $D$ is taken as the nonergodicity limit; (b) Phase diagram showing $\beta A$ as a function of $\varphi$, extracted by taking the iso-$D$ lines from (a). The dashed line signals the onset of the glass region; state points with the same color coding have the same value of diffusion coefficient.

Figure 5

Dependence of the 2D Hertzian strength $A$ on the cross-linker concentration for microgels of various sizes. Values of $\beta A$ are extracted via the theoretical calculation of $Y$ and ${\sigma }_{\mathrm{ext}}$ for microgels assembled with $N\approx 2000$ (circles), 3000 (squares), and 5000 (triangles) monomers for $c=3\%$ (orange), 5% (red), and 10% (dark red). Symbols are slightly displaced on the $x$ axis to enhance readability. The dashed line indicates the approximate largest value of the Hertzian strength for which a reentrant transition could be observed (see Fig. 4). Data are averaged over four different topologies for each combination of $N$ and $c$.