^{1}, D. Schaar

^{1}, H. G. E. Hentschel

^{1}, J. Hay

^{1}, Piotr Habdas

^{2}and Eric R. Weeks

^{1}

### Abstract

We examine the response of a dense colloidal suspension to a local force applied by a small magnetic bead. For small forces, we find a linear relationship between the force and the displacement, suggesting the medium is elastic, even though our colloidal samples macroscopically behave as fluids. We interpret this as a measure of the strength of colloidal caging, reflecting the proximity of the samples' volume fractions to the colloidal glass transition. The strain field of the colloidal particles surrounding the magnetic probe appears similar to that of an isotropic homogeneous elastic medium. When the applied force is removed, the strain relaxes as a stretched exponential in time. We introduce a model that suggests this behavior is due to the diffusive relaxation of strain in the colloidal sample.

We thank R. E. Courtland, S. A. Koehler, M. Fuchs, K. S. Schweizer, and M. Wyart for helpful discussions. We thank A. Schofield for providing our colloidal samples. The work of D.A., D.S., P.H., and J.H. was supported by National Aeronautics and Space Administration (NASA) (Grant No. NAG3-2284). The work of E.R.W. was supported by National Science Foundation (NSF) (Grant No. CHE-0910707).

I. INTRODUCTION

II. EXPERIMENTAL METHODS

III. LINEAR ELASTIC RESPONSE

IV. PARTICLE DISPLACEMENT FIELDS

V. DECAY OF STRAIN

A. Experimental observations

B. Model of relaxing bead

VI. CONCLUSIONS

### Key Topics

- Colloidal systems
- 46.0
- Elasticity
- 21.0
- Glass transitions
- 18.0
- Elastic moduli
- 14.0
- Viscosity
- 13.0

##### B01J13/00

## Figures

The solid line indicates the applied force as a function of time. The points show the measured displacement Δ*x* of the magnetic bead. (Inset) Same data plotted as Δ*x* against *F*. The dashed line is a fit to the data, with the slope leading to an effective spring constant *k* = 6.8 ± 0.1 pN/μm, offset vertically for clarity. The arrows indicate locations where the force was held constant. The volume fraction is ϕ = 0.55.

The solid line indicates the applied force as a function of time. The points show the measured displacement Δ*x* of the magnetic bead. (Inset) Same data plotted as Δ*x* against *F*. The dashed line is a fit to the data, with the slope leading to an effective spring constant *k* = 6.8 ± 0.1 pN/μm, offset vertically for clarity. The arrows indicate locations where the force was held constant. The volume fraction is ϕ = 0.55.

Applied force as a function of time, for the three largest maximum forces (see Table I ). For all curves, *F*(*t*) = 0 for *t* < 0 s.

Applied force as a function of time, for the three largest maximum forces (see Table I ). For all curves, *F*(*t*) = 0 for *t* < 0 s.

(a) Raw image of particles, before the force is applied. (b) Difference between “before” and “after” a single force pulse is applied. (c) Difference between two “after” images for two subsequent pulses. (d) As the images are reproducible, a sequence of eight “before” pictures is averaged together, and likewise eight “after” pictures. This picture is the difference between these average images. For all pictures, the scale bar is 10 microns long, the volume fraction is ϕ = 0.49, and the applied force is *F* _{max} = 0.29 nN.

(a) Raw image of particles, before the force is applied. (b) Difference between “before” and “after” a single force pulse is applied. (c) Difference between two “after” images for two subsequent pulses. (d) As the images are reproducible, a sequence of eight “before” pictures is averaged together, and likewise eight “after” pictures. This picture is the difference between these average images. For all pictures, the scale bar is 10 microns long, the volume fraction is ϕ = 0.49, and the applied force is *F* _{max} = 0.29 nN.

(a) Displacement field based on data shown in Fig. 3(b) . The arrows indicate displacements of the colloidal particles. (b) Residual displacement field after subtracting off the fit to Eq. (2) . The arrows are magnified by a factor of 5; in reality, the longest displacement vectors in panel (b) are 0.3 μm. The central region near the magnetic bead is removed for clarity. For both panels, the circles indicate the initial and final positions of the magnetic bead, which moved from right to left, and are drawn to scale. The scale bar is 10 microns long. The data correspond to Figs. 4 and 5 : ϕ = 0.49 and *F* _{max} = 0.29 nN. Note that a displacement vector is calculated for every pixel in the raw images; here, only every 6th vector is drawn.

(a) Displacement field based on data shown in Fig. 3(b) . The arrows indicate displacements of the colloidal particles. (b) Residual displacement field after subtracting off the fit to Eq. (2) . The arrows are magnified by a factor of 5; in reality, the longest displacement vectors in panel (b) are 0.3 μm. The central region near the magnetic bead is removed for clarity. For both panels, the circles indicate the initial and final positions of the magnetic bead, which moved from right to left, and are drawn to scale. The scale bar is 10 microns long. The data correspond to Figs. 4 and 5 : ϕ = 0.49 and *F* _{max} = 0.29 nN. Note that a displacement vector is calculated for every pixel in the raw images; here, only every 6th vector is drawn.

Rescaled displacement vectors as a function of θ; compare with Eq. (2) . The points are the data and the solid line is the fit to the equation. The data correspond to Figs. 3(b) and 4(a) , using only data with *r* > *r* _{0} = *a* _{MB}. For this fit, σ was constrained to be 1/2.

Rescaled displacement vectors as a function of θ; compare with Eq. (2) . The points are the data and the solid line is the fit to the equation. The data correspond to Figs. 3(b) and 4(a) , using only data with *r* > *r* _{0} = *a* _{MB}. For this fit, σ was constrained to be 1/2.

Young's modulus *E* as a function of volume fraction ϕ. Due to an inadequately defined applied force, *E* is overestimated although this affects all points equally (by a multiplicative factor) and does not change the shape of the curve; see the text for a discussion. The inset shows the same data plotted as a function of ϕ_{ c } − ϕ with ϕ_{ c } = 0.64. The lines in the main plot and the inset are the fit to the data using *E* = *E* _{0}(ϕ_{ c } − ϕ)^{−β} with *E* _{0} = 0.4 Pa and β = 1.84 ± 0.40. The symbol size indicates the uncertainty.

Young's modulus *E* as a function of volume fraction ϕ. Due to an inadequately defined applied force, *E* is overestimated although this affects all points equally (by a multiplicative factor) and does not change the shape of the curve; see the text for a discussion. The inset shows the same data plotted as a function of ϕ_{ c } − ϕ with ϕ_{ c } = 0.64. The lines in the main plot and the inset are the fit to the data using *E* = *E* _{0}(ϕ_{ c } − ϕ)^{−β} with *E* _{0} = 0.4 Pa and β = 1.84 ± 0.40. The symbol size indicates the uncertainty.

Plots of the displacement of the magnetic bead as a function of time, after the force is removed, for ϕ = 0.49. Panel (a) shows a linear-linear plot and panel (b) shows a log-linear plot. The values of *F* _{max} are given in Table I , with the largest initial displacement (red squares) corresponding to the largest force and the smallest initial displacement (purple pluses) corresponding to the smallest force. In (b), lines are fit to the initial data (*t* < 0.5 s) indicating decay time constants of 0.47 s, 0.38 s, 0.37 s, 0.31 s, and 0.46 s (from largest *F* _{max} to smallest).

Plots of the displacement of the magnetic bead as a function of time, after the force is removed, for ϕ = 0.49. Panel (a) shows a linear-linear plot and panel (b) shows a log-linear plot. The values of *F* _{max} are given in Table I , with the largest initial displacement (red squares) corresponding to the largest force and the smallest initial displacement (purple pluses) corresponding to the smallest force. In (b), lines are fit to the initial data (*t* < 0.5 s) indicating decay time constants of 0.47 s, 0.38 s, 0.37 s, 0.31 s, and 0.46 s (from largest *F* _{max} to smallest).

Relaxation curves for several experiments demonstrating that the decay is faster for samples with higher ϕ. (a) Comparison of two samples with ϕ as indicated that have nearly the same initial displacement. For the ϕ = 0.44 data, the force is *F* _{max} = 0.13 nN, and for the ϕ = 0.49 data, the force is *F* _{max} = 0.29 nN. Two different instances are shown for the ϕ = 0.49 data (triangles and pluses). (b) Comparison of three samples with the same force (*F* _{max} = 0.29 nN) but different ϕ as indicated.

Relaxation curves for several experiments demonstrating that the decay is faster for samples with higher ϕ. (a) Comparison of two samples with ϕ as indicated that have nearly the same initial displacement. For the ϕ = 0.44 data, the force is *F* _{max} = 0.13 nN, and for the ϕ = 0.49 data, the force is *F* _{max} = 0.29 nN. Two different instances are shown for the ϕ = 0.49 data (triangles and pluses). (b) Comparison of three samples with the same force (*F* _{max} = 0.29 nN) but different ϕ as indicated.

Displacement plotted as a function of for (a) ϕ = 0.47 and (b) ϕ = 0.49. The different symbols indicate different values of *F* _{max}. The values of *F* _{max} are given in Table I , with the largest initial displacement (red squares) corresponding to the largest force and the smallest initial displacement (purple pluses) corresponding to the smallest force. The straight lines indicate fits to ∼ . For (a), the values of *t* _{0} are 0.50, 0.29, and 0.31 s (top to bottom). For (b), the values of *t* _{0} are 0.23, 0.21, and 0.19 s (top to middle).

Displacement plotted as a function of for (a) ϕ = 0.47 and (b) ϕ = 0.49. The different symbols indicate different values of *F* _{max}. The values of *F* _{max} are given in Table I , with the largest initial displacement (red squares) corresponding to the largest force and the smallest initial displacement (purple pluses) corresponding to the smallest force. The straight lines indicate fits to ∼ . For (a), the values of *t* _{0} are 0.50, 0.29, and 0.31 s (top to bottom). For (b), the values of *t* _{0} are 0.23, 0.21, and 0.19 s (top to middle).

## Tables

The five different maximum forces applied, and the integrated impulse *I* = ∫*F*(*t*)*dt*. The calibration procedure (described in the text) has an intrinsic *F* _{max} uncertainty of ±0.05 nN and an *I* uncertainty of ±0.005 nN s. Due to variability between different magnetic beads, for a given magnetic bead there is also an overall systematic uncertainty of ±10%. Graphs of *F*(*t*) for the three largest values of *F* _{max} are shown in Fig. 2 .

The five different maximum forces applied, and the integrated impulse *I* = ∫*F*(*t*)*dt*. The calibration procedure (described in the text) has an intrinsic *F* _{max} uncertainty of ±0.05 nN and an *I* uncertainty of ±0.005 nN s. Due to variability between different magnetic beads, for a given magnetic bead there is also an overall systematic uncertainty of ±10%. Graphs of *F*(*t*) for the three largest values of *F* _{max} are shown in Fig. 2 .

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