Interpenetration of polymeric microgels at ultrahigh densities

Abstract

Soft particles such as polymeric microgels can form ultra-dense phases, where the average center-to-center distance a _s can be smaller than the initial unperturbed particle diameter σ ₀, due to their ability to interpenetrate and compress. However, despite of the effort devoted to microgels at ultrahigh densities, we know surprisingly little about their response to their environment at effective volume fractions ϕ _eff above close packing (ϕ _cp ), and the existing information is often contradictory. Here we report direct measurements of the size and shape of poly(N-isopropylacrylamide) microgels at concentrations below and above ϕ _cp using the zero average contrast method in small-angle neutron scattering. We complement these experiments with measurements of the average interparticle distances using small-angle x-ray scattering, and a determination of the glass transition using dynamic light scattering. This allows us to unambiguously decouple interaction effects from density-dependent variations of the particle size and shape at all values of ϕ _eff . We demonstrate that the microgels used in this study significantly interpenetrate and thus change their size and shape only marginally even for ϕ _eff ≫ ϕ _cp , a finding that may require changes in the interpretation of a number of previously published studies on the structural and dynamic properties of dense soft particle systems.

Figure 1

Swelling behavior of microgels. Temperature dependence of the hydrodynamic radius R _h for the deuterated and hydrogenated particles in H₂O. Shown are the actual values (A) as well as rescaled values (B) where the temperature-dependent values R _h(T) are normalized by R _h(15 °C) as a function of temperature for deuterated (squares) and hydrogenated (circles) particles. Also shown are the corresponding values obtained in the ZAC solvent for Hm (open black circles) and Dm (open red squares), as well as the results obtained in 12:88 H₂O:D₂O, (Hm: open black crossed circles, Dm: open red crossed squares). Error bars are calculated based on the standard deviation of the measurements of the decay time for the individual scattering angles, and the resulting statistical error in the calculation of the average hydrodynamic radius from the q ²-dependence of these average decay times as described in the methods section. Note that for most data points the resulting error bars are smaller than the symbol size.

Figure 2

Small-angle neutron and X-ray scattering experiments. Shown are examples of the scattered intensity I(q) as a function of the scattering vector q for a 50–50 mixture of Hm and Dm particles at T = 16.4 °C in a ZAC solvent mixture for different values of the effective volume fraction 0.34 ≤ ϕ _eff ≤ 1.68 (A). Note that the data are in absolute units (cm⁻¹), but the individual scattering curves are shifted in order to improve their visibility (shift factors: 1, 1.8, 4, 8 and 18 for increasing concentrations). The solid lines are the fits of the form factor for fuzzy spheres (see SI for details) to the experimental data. Also shown is the normalised scattering intensity $I\left(q\right)/C~\frac{d\sigma }{d\mathrm{\Omega }}\left(q\right)$ as a function of the scattering vector q for the same samples (inset), where C is the weight concentration. Examples of the scattered intensity I(q) as a function of the scattering vector q from SAXS experiments with a 50–50 mixture of Hm and Dm particles at T = 15 °C in a ZAC solvent mixture for different values of the effective volume fraction 0.34 ≤ ϕ _eff ≤ 1.68, bottom to top (B). The position q ^* of the maximum of the resulting structure factor peak is also indicated. The data is offset along the ordinate for clarity.

Figure 3

Summary of the results obtained by SANS and SAXS. Shown are: Normalised overall dimensions R _SANS/R _SANS,0 as a function of ϕ _eff obtained from SANS measurements using ZAC conditions at T = 15 °C (ILL) and 16.4 °C (PSI), respectively, where R _SANS,0 corresponds to the value at ϕ _eff → 0 (solid black circles). The values correspond to the average obtained in different independent measurement campaigns at two different neutron sources, and the error bars reflect the standard deviations of these experiments. Also shown are the overall dimensions R _SANS/R _SANS,0 as a function of ϕ _eff at T = 27.1 °C (ILL) and 27.4 °C (PSI) (solid black triangles). The average normalised center-center distances a _s/a ₀ from SAXS are shown as the red solid circles for T = 15 °C and triangles for T = 27 °C, where a ₀ = 2R _h[T] × R _SANS/R _SANS,0. The average normalised center-center distances a _s/a ₀ from CLSM are shown as the red solid squares for T = 15 °C. The red solid line represents the dependence of the normalised center-center distance on the number density for particles interacting via a soft repulsive potential given by $a_s/a_0~{\phi }_{eff}^{-1/3}$ expected for ϕ _eff  ≥ ϕ _rcp, where ϕ _rcp corresponds to random close packing. The location of the glass transition is indicated by the black solid arrow (T = 15 °C) and the black dashed arrow (T = 27 °C) at the bottom, and an estimate of the volume fraction ϕ _eff where the outer shell touches the dense core of the nearest neighbour particles is given as the short solid black (T = 15 °C) and dashed (T = 27 °C) lines at the top.

Figure 4

Dynamics of dense suspensions. (A) Normalized field autocorrelation functions g ₁(t) as a function of delay time t for 50–50 particle mixtures in ZAC solvent at T = 15 °C and effective volume fractions ϕ _eff = 0.17, 0.34, 0.50, 0.59, 0.67, 0.84, from left to right. (B) Normalized field autocorrelation functions g ₁(t) as a function of delay time t for 50–50 particle mixtures in ZAC solvent at T = 27 °C and effective volume fractions ϕ _eff = 0.12, 0.24, 0.36, 0.42, 0.48, 0.54, 0.6, 0.96, from left to right. (C) Normalized slow relaxation time τ _s/τ ₀ as a function of ϕ _eff at T = 15 °C. (D) Normalized slow relaxation time τ _s/τ ₀ as a function of ϕ _eff at T = 27 °C. Also shown are fitted curves τ _s/τ ₀ = (1 − ϕ _eff/ϕ _g)^−α as solid lines, where α = 1.6 for T = 15 °C and α = 1.2 for T = 27 °C, and the resulting estimate of the glass transition volume fraction ϕ _g as the dotted lines.

Figure 5

Assessing the sensitivity of the SANS analysis. Comparison of the concentration-normalized scattered SANS intensity I(q)/C, where C is the weight concentration, in ZAC contrast from samples with an effective volume fraction ϕ _eff = 0.67 and ϕ _eff = 1.0 in a residual plot [I(ϕ _eff = 1.0)/C − I(ϕ _eff = 0.67)/C]/I(ϕ _eff = 0.67/C). Also shown are calculated form factors for different values of the effective polydispersity also as residual plots given by [P _fit(ϕ _eff,2, PD ₂) − P _fit(ϕ _eff,1, PD ₁)]/P _fit(ϕ _eff,1, PD ₁), where PD ₁ = 12% and PD ₂ = 17% (red line) or 22% (blue line), respectively.

Figure 6

Slow dissolution of jammed samples. Upper panel: Dissolution of a small droplet of a microgel sample equilibrated at T = 15 °C upon injection into a water bath at T = 30 °C. Top row: Initial sample concentration ϕ _eff = 0.59. Middle row: Initial sample concentration ϕ _eff = 0.73. Bottom row: Initial sample concentration ϕ _eff = 1.17. Pictures shown were taken at different times after injecting the droplet. Scale bars correspond to 10 mm. Lower panel: Schematic description of the response of a microgel dispersion to a rapid temperature jump from 15 °C (T ≪ T _VPT) to 30 °C (T ≲ T _VPT) at three different volume fractions ϕ _eff < ϕ _g (left), ϕ _g < ϕ _eff < ϕ _cp (middle), and ϕ _eff > ϕ _cp (right), respectively, where ϕ _cp stands for the volume fraction where particles are close packed or jammed and start to significantly interpenetrate. Also shown is a magnified view of the temporary entanglements at ϕ _eff > ϕ _cp (right), which become buried and trapped after the temperature jump.