A new look at effective interactions between microgel particles

Abstract

Thermoresponsive microgels find widespread use as colloidal model systems, because their temperature-dependent size allows facile tuning of their volume fraction in situ. However, an interaction potential unifying their behavior across the entire phase diagram is sorely lacking. Here we investigate microgel suspensions in the fluid regime at different volume fractions and temperatures, and in the presence of another population of small microgels, combining confocal microscopy experiments and numerical simulations. We find that effective interactions between microgels are clearly temperature dependent. In addition, microgel mixtures possess an enhanced stability compared to hard colloid mixtures - a property not predicted by a simple Hertzian model. Based on numerical calculations we propose a multi-Hertzian model, which reproduces the experimental behavior for all studied conditions. Our findings highlight that effective interactions between microgels are much more complex than usually assumed, displaying a crucial dependence on temperature and on the internal core-corona architecture of the particles.

Fig. 1

Temperature-dependent structure, dynamics and interaction potential of one-component microgel suspensions. Symbols indicate experimental data, solid lines represent simulations. Color legend applies to all panels. a Radial distribution functions g(r) from simulations and experiments. Panels show data for ${\varphi }_{\mathrm{eff,c}}=0.26,0.37$ and 0.49 (left to right, evaluated at T=15°C) for 15°C ≤ T ≤ 30 °C. Graphs are offset along the y-axis for clarity. Downward pointing arrows indicate the hydrodynamic diameter of colloids at each T. b MSDs for 15 °C ≤ T ≤ 30 °C reconstructed from the x,y trajectories, i.e. 〈x² + y²〉, with ${\varphi }_{\mathrm{eff,c}}=0.26,0.37$ and 0.49 (from top to bottom). c Hertzian interaction potential at different temperatures: U_cc = 400, 520, 640, 760k_BT for T = 15, 20, 25, 30 °C respectively. The distance is rescaled by σ_eff = 2R_H

Fig. 2

Structure and dynamics for two state points at different temperature with equivalent packing fraction. (a) Experimental $g\left(r\right)$ and (b) MSD for samples with ϕ_eff,c = 0.37 at two different temperatures (T = 15 and 25 °C) corrected for size. D₀ is the zero-colloid limit diffusion coefficient and its associated MSD (dashed line) is also shown

Fig. 3

Experimental and numerical structural correlations for all investigated binary mixtures. Experimental g(r)s (colored squares) are compared to numerical ones (solid lines) based on the multi-Hertzian model. Data for different samples are offset in y for clarity. The color legend applies to the entire graph. Values of ϕ_eff,c and ϕ_eff,d at 15 °C are given for each row and column, respectively. For higher temperatures, the values of ϕ_eff,c, ϕ_eff,d can be found in Table 1

Fig. 4

Experimental and numerical mean square displacements for all investigated state points. Diamonds denote 2D experimental data (〈x² + y²〉), while solid lines represent the corresponding simulation results based on the MH model. The color legend applies to the entire graph. Values of ϕ_eff,c and ϕ_eff,d at 15 °C are given for each row and column, respectively. For higher temperatures, the values of ϕ_eff,c,ϕ_eff,d can be found in Table 1

Fig. 5

The multi-Hertzian model. a Calculated effective potential between two microgels as a function of their center-to-center distance. Lines are fits to three different Hertzian contributions, labeled respectively as Hertzian 1, which corresponds to the calculated elastic moduli and whose effective diameter σ is used to rescale the x-axis, Hertzian 2 and Hertzian 3 representing the contributions of the inner structure of the microgels. For the reported microgel, the fitted strengths are U₁ = 335k_BT, U₂ = 1182k_BT and U₃ = 2617k_BT and the fitted lengths are σ₁ = σ, σ₂ = 0.92σ and σ₃ = 0.8354σ in good qualitative agreement with the ones used to fit experimental data whose parameters are given in Table 2; b the model describing experimental data with the employed interactions lengths: σ_core, below which core-core interactions take place, σ_mid relevant to the onset of core-corona interactions and σ_corona which reflect the heterogeneous nature of the outer corona shell. A comparison with the Hertzian model is also provided. Note the logarithmic scales on the y-axis for both panels

Fig. 6

Typical $V_{\mathrm{eff,cc}}^{\mathrm{MH}}$ and resultant g(r)s for one state point at different temperatures. a Calculated effective potential $\beta V_{\mathrm{eff,cc}}^{\mathrm{MH}}$ based on temperature-dependent $V_{cc}^{\mathrm{MH}}$ and V_cd. At T = 15 °C, $\beta V_{\mathrm{eff,cc}}^{\mathrm{MH}}$ displays a shallow negative minimum at $~-0.6k_{\mathrm{B}}T$. (b) Comparison between numerical g(r)s (solid lines) and experimental g(r)s (colored squares). From top to bottom temperature increases from 15 to 30 °C. Downward pointing arrows indicate the hydrodynamic diameter of colloids which does not generally coincide with the first peak position at any temperature