^{1,2,a)}, Luna Imperiali

^{2,3}, Jin-Woong Kim

^{4}, Alberto Fernández-Nieves

^{5}and David A. Weitz

^{6}

### Abstract

A direct consequence of the finite compressibility of a swollen microgel is that it can shrink and deform in response to an external perturbation. As a result, concentrated suspensions of these particles exhibit relaxation dynamics and rheological properties which can be very different with respect to those of a hard sphere suspension or an emulsion. We study the reduction in size of ionic microgels in response to increasing number of particles to show that particle shrinkage originates primarily from steric compression, and that the effect of ion-induced de-swelling of the polymer network is negligible. With increasing particle concentration, the single particle dynamics switch from those typical of a liquid to those of a super-cooled liquid and finally to those of a glass. However, the transitions occur at volume fractions much higher than those characterizing a hard sphere system. In the super-cooled state, the distribution of displacements is non-Gaussian and the dependence of the structural relaxation time on volume fraction is describable by a Volger-Fulcher-Tammann function.

A.F-N. thanks the Government of Andalucia (Project No. P07-FQM-03116), the FEDER program and the University of Almeria. The work at Harvard was supported by the NSF (Grant No. DMR-1006546) and the Harvard MRSEC (Grant No. DMR-0820484).

I. INTRODUCTION

II. MATERIALS AND CHARACTERIZATION METHODS

A. Particle synthesis and characterization

B. Particle characterization: Viscometry and titration measurements

C. Dynamic light scattering

D. Static light scattering

E. Confocal measurements of particle radius and dynamics

III. RESULTS AND DISCUSSION

A. Origin of particle de-swelling

B. Dynamics in dense suspensions

IV. CONCLUSIONS

### Key Topics

- Microgels
- 59.0
- Polymers
- 30.0
- Suspensions
- 29.0
- Light scattering
- 14.0
- Relaxation times
- 11.0

## Figures

Dependence of the relative viscosity η_{ r } = η/η_{ o } on polymer concentration *c* at *p*H7 (circles) and *p*H8 (squares). The solid line is a fit to the Einstein-Batchelor relation: η_{ r } = 1 + 2.5(*k*·*c*) + *B*(*k*·*c*)^{2} giving *k* = (2.2 ± 0.2) × 10^{2} and *B* = 4 ± 2. Dashed line is a fit to the Einstein equation. The inset shows the fluorescent particles deposited on a glass slide.

Dependence of the relative viscosity η_{ r } = η/η_{ o } on polymer concentration *c* at *p*H7 (circles) and *p*H8 (squares). The solid line is a fit to the Einstein-Batchelor relation: η_{ r } = 1 + 2.5(*k*·*c*) + *B*(*k*·*c*)^{2} giving *k* = (2.2 ± 0.2) × 10^{2} and *B* = 4 ± 2. Dashed line is a fit to the Einstein equation. The inset shows the fluorescent particles deposited on a glass slide.

Number of ionized groups per particle *q* as function of *p*H, obtained from titration of three solutions at polymer concentrations c= 0.19% (diamonds), 0.56% (squares), 0.9% (triangles). The solid line is a fit to the equation and gives *Q* = 6.7 × 10^{6} and *pK* _{ a } = 5. The inset shows the product of the total number of ions per unit volume and particle mass as function of polymer concentration *c* for samples at *p*H ≃ 7; the slope of the linear fit is *Q* = 7.8 × 10^{6}.

Number of ionized groups per particle *q* as function of *p*H, obtained from titration of three solutions at polymer concentrations c= 0.19% (diamonds), 0.56% (squares), 0.9% (triangles). The solid line is a fit to the equation and gives *Q* = 6.7 × 10^{6} and *pK* _{ a } = 5. The inset shows the product of the total number of ions per unit volume and particle mass as function of polymer concentration *c* for samples at *p*H ≃ 7; the slope of the linear fit is *Q* = 7.8 × 10^{6}.

MSD curves for different wavevectors *q*(1/μm). *q* = 9 (squares), 13.2 (circles), 17 (triangles), 20.2 (reverse triangles), 22.9 (diamonds) as a function of lag-time τ for a sample at ζ ≃ 1.2. In the inset, the time average correlation function of the electric field is reported as function of lag-time. Symbols are the same of the main plot.

MSD curves for different wavevectors *q*(1/μm). *q* = 9 (squares), 13.2 (circles), 17 (triangles), 20.2 (reverse triangles), 22.9 (diamonds) as a function of lag-time τ for a sample at ζ ≃ 1.2. In the inset, the time average correlation function of the electric field is reported as function of lag-time. Symbols are the same of the main plot.

Scattered intensity (arbitrary units) as function of wavevector *q* for samples at volume fractions ζ = 0.15 (open circles) and ζ = 3.1 (solid circles). Solid line is the fitting of data-points of the dilute sample to the form factors of polydisperse inhomogeneous spheres (Eq. (9)) giving a radius *R* = 0.65 μm. Dashed line is the form factor of monodisperse inhomogeneous spheres of radius *R* = 0.48 μm.

Scattered intensity (arbitrary units) as function of wavevector *q* for samples at volume fractions ζ = 0.15 (open circles) and ζ = 3.1 (solid circles). Solid line is the fitting of data-points of the dilute sample to the form factors of polydisperse inhomogeneous spheres (Eq. (9)) giving a radius *R* = 0.65 μm. Dashed line is the form factor of monodisperse inhomogeneous spheres of radius *R* = 0.48 μm.

Left: 2D superposition of 25 slices vertically separated by 0.3 μm obtained from a solution of microgels at *c* = 0.6% (ζ = 1.35). Right: Contours of the only particles measured by the software after image processing.

Left: 2D superposition of 25 slices vertically separated by 0.3 μm obtained from a solution of microgels at *c* = 0.6% (ζ = 1.35). Right: Contours of the only particles measured by the software after image processing.

Prediction of the dependence of the swelling ratio α, normalized to its value at infinite dilution α_{∞}, on the generalized volume fraction ζ, as obtained from Eq. (11) (dashed line), and from Eq. (17) (solid line).

Prediction of the dependence of the swelling ratio α, normalized to its value at infinite dilution α_{∞}, on the generalized volume fraction ζ, as obtained from Eq. (11) (dashed line), and from Eq. (17) (solid line).

Dependence of the microgel radius on generalized volume fraction. The radius measured from confocal images *R* _{ conf } is reported on the left axis for samples at *p*H7 (circles) and *p*H8 (stars). The radius measured from static light scattering *R* _{ SLS } (crosses) is reported on the right axis for samples at *p*H7. Vertical axes are scaled to collapse *R* _{ conf } and *R* _{ SLS } in dilute samples on the same horizontal line (dashed line). Solid line is a plot of the equation *R*∝ζ^{−1/3}.

Dependence of the microgel radius on generalized volume fraction. The radius measured from confocal images *R* _{ conf } is reported on the left axis for samples at *p*H7 (circles) and *p*H8 (stars). The radius measured from static light scattering *R* _{ SLS } (crosses) is reported on the right axis for samples at *p*H7. Vertical axes are scaled to collapse *R* _{ conf } and *R* _{ SLS } in dilute samples on the same horizontal line (dashed line). Solid line is a plot of the equation *R*∝ζ^{−1/3}.

Left: Dependence of the MSD on lag-delay time τ for particle volume fractions ζ = 0.02 (squares); 0.5 (circles); 1.22 (triangles); 1.96 (diamonds); 2.77 (stars). Open symbols are from DLS and solid symbols from CM. The images on the right represent particle trajectories at ζ = 0.02 (top) and ζ = 1.96 (bottom) of duration 26 and 500 s respectively.

Left: Dependence of the MSD on lag-delay time τ for particle volume fractions ζ = 0.02 (squares); 0.5 (circles); 1.22 (triangles); 1.96 (diamonds); 2.77 (stars). Open symbols are from DLS and solid symbols from CM. The images on the right represent particle trajectories at ζ = 0.02 (top) and ζ = 1.96 (bottom) of duration 26 and 500 s respectively.

Left: displacement distributions for samples at ζ = 0.02 (top) and ζ = 1.96 (bottom). Lines are Gaussian fits to the data. For each sample, solid and empty symbols are for shorter and longer τ values respectively. Right: Non-Gaussian parameter α_{2} for samples at different ζ as function of lag-time. Symbols are the same as in Figure 8.

Left: displacement distributions for samples at ζ = 0.02 (top) and ζ = 1.96 (bottom). Lines are Gaussian fits to the data. For each sample, solid and empty symbols are for shorter and longer τ values respectively. Right: Non-Gaussian parameter α_{2} for samples at different ζ as function of lag-time. Symbols are the same as in Figure 8.

Structural relaxation time τ_{ r } for microgel solutions at *p*H7 (circles) and *p*H8 (diamonds) and relative zero-shear viscosity (squares) as function of ζ. Solid line is a fit of the data in the supercooled state (ζ > 1.1) to the VFT equation with *A* = 22.32 and ζ_{ g } = 6.87. The inset shows a log-log plot of ln (τ_{ r }) as function of the normalized volume fraction ζ/ζ_{ g }. Solid line is the same as in the main figure and the dashed line describes the divergence of hard spheres according to with *C* = 0.0098 and ϕ_{ g } = 0.64 (Ref. 35).

Structural relaxation time τ_{ r } for microgel solutions at *p*H7 (circles) and *p*H8 (diamonds) and relative zero-shear viscosity (squares) as function of ζ. Solid line is a fit of the data in the supercooled state (ζ > 1.1) to the VFT equation with *A* = 22.32 and ζ_{ g } = 6.87. The inset shows a log-log plot of ln (τ_{ r }) as function of the normalized volume fraction ζ/ζ_{ g }. Solid line is the same as in the main figure and the dashed line describes the divergence of hard spheres according to with *C* = 0.0098 and ϕ_{ g } = 0.64 (Ref. 35).

## Tables

Physical and chemical properties of the partially ionic microgel particles: polymer volume fraction in a particle φ_{0} at preparation conditions; Flory interaction parameter χ at 20 °C; particle volume at preparation conditions *v* _{0}; particle volume in the swollen state *v*; swelling ratio at infinite dilution α_{∞}; particle polymer mass *m* _{ p }; polymer charge *Q* per particle; effective number of chains *N* _{ c }; counterion screening length ; Γ is the fraction of counterions that leave the microgel particle at infinite dilution.

Physical and chemical properties of the partially ionic microgel particles: polymer volume fraction in a particle φ_{0} at preparation conditions; Flory interaction parameter χ at 20 °C; particle volume at preparation conditions *v* _{0}; particle volume in the swollen state *v*; swelling ratio at infinite dilution α_{∞}; particle polymer mass *m* _{ p }; polymer charge *Q* per particle; effective number of chains *N* _{ c }; counterion screening length ; Γ is the fraction of counterions that leave the microgel particle at infinite dilution.

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