^{1,a)}, Robert Botet

^{2}, Martine Meireles

^{3}, Günter K. Auernhammer

^{4}and Bernard Cabane

^{5}

### Abstract

The compressive yield stress of particle gels shows a highly nonlinear dependence on the packing fraction. We have studied continuous compression processes and discussed the packing-fraction dependence with the particle-scale rearrangements. The two-dimensional simulation of uniaxial compression was applied to fractal networks, and the required compressive stresses were evaluated for a wide range of packing fractions that approached close packing. The compression acts to reduce the size of the characteristic structural entities (i.e., the correlation length of the structure). We observed three stages of compression: (I) Elastic-dominant regime; (II) single-mode plastic regime, where the network strengths are determined by the typical length scale and the rolling mode; and (III) multimode plastic regime, where sliding mode and connection breaks are important. We also investigated the way of losing the fractal correlation under compression. It turns out that both fractal dimension and correlation length ξ start to change from the early stage of compression, which is different from the usual assumption in theoretical models.

The authors would like to thank Professor Morton Denn for valuable suggestions on the manuscript, and R.S. would like to express his gratitude to Professor Richard Buscall for fruitful discussions, valuable suggestions, and warm encouragement. This work was supported by Project ANR-BLAN06-3-144.

I. INTRODUCTION

II. METHOD

A. Initial configuration: Fractalgel

B. Contact forces

C. Equation of motion

D. Uniaxial compression and static equilibrium

E. Non real-time simulation for quasistatic simulation

III. RESULTS AND DISCUSSION

A. Parameters for contact model

B. Parameters of the compression simulation

C. Uniformity of compression

D. Data of compressive consolidation

1. Bond creation

2. Stored and dissipated energies

3. Bond rupture

E. Three stages of compression

1. Elastic-dominant regime

2. Single-mode plastic regime

3. Multimode plastic regime

F. Fractal correlation under compression

IV. CONCLUSION

### Key Topics

- Fractals
- 48.0
- Gels
- 44.0
- Shear deformation
- 17.0
- Yield stress
- 10.0
- Colloidal systems
- 8.0

## Figures

The density-density correlation functions were evaluated with the fractal networks of prepared in the box . The averages and standard deviations were taken over ten samples. The red line indicates the slope corresponding to .

The density-density correlation functions were evaluated with the fractal networks of prepared in the box . The averages and standard deviations were taken over ten samples. The red line indicates the slope corresponding to .

The three-point bending test for a linear aggregate consists of the three steps i–iii. When a smaller external force ( ) is applied, the elastic deformation is seen (a), and when it is slightly larger ( ), the bond breakups cause the plastic deformation (b).

The three-point bending test for a linear aggregate consists of the three steps i–iii. When a smaller external force ( ) is applied, the elastic deformation is seen (a), and when it is slightly larger ( ), the bond breakups cause the plastic deformation (b).

The y dependence of packing fractions is shown for every second compression step ( , i = 1, 3, 5…). The solid line indicates the initial configuration.

The y dependence of packing fractions is shown for every second compression step ( , i = 1, 3, 5…). The solid line indicates the initial configuration.

The compressive yield stress is evaluated by the simulations of stepwise compression. Its averages and standard deviations taken over ten runs are shown. The red squares represent the result of the bond 1 simulations, and the blue circles of the bond 2 simulations. The dotted curve shows the fitting function of Eq. (11) , and (solid and dashed) straight lines show the power-law functions (12) . The arrows indicate ranges of the three regimes: (I) Elastic-dominant regime, (II) single-mode plastic regime, and (III) multimode plastic regime. The average exponents λ are written on the figure.

The compressive yield stress is evaluated by the simulations of stepwise compression. Its averages and standard deviations taken over ten runs are shown. The red squares represent the result of the bond 1 simulations, and the blue circles of the bond 2 simulations. The dotted curve shows the fitting function of Eq. (11) , and (solid and dashed) straight lines show the power-law functions (12) . The arrows indicate ranges of the three regimes: (I) Elastic-dominant regime, (II) single-mode plastic regime, and (III) multimode plastic regime. The average exponents λ are written on the figure.

(a) The mean contact number per particles increases as the compression proceeds. The standard deviations are taken over the averages of ten simulations. (b) The successive increments of mean bond stored and mean dissipated energy rates are shown, where . The rates are obtained by normalizing with the compressive strain . The averages and standard deviations are taken over ten runs of the bond 1 simulation.

(a) The mean contact number per particles increases as the compression proceeds. The standard deviations are taken over the averages of ten simulations. (b) The successive increments of mean bond stored and mean dissipated energy rates are shown, where . The rates are obtained by normalizing with the compressive strain . The averages and standard deviations are taken over ten runs of the bond 1 simulation.

Bond rupture rates are compared for two types of bond: (a) Bond 1 and (b) bond 2. The bond ruptures are recorded by the causing stress. The rates of the separation due to normal forces are plotted by squares, the ones of the regeneration due to sliding forces or rolling moments by triangles and circles, respectively.

Bond rupture rates are compared for two types of bond: (a) Bond 1 and (b) bond 2. The bond ruptures are recorded by the causing stress. The rates of the separation due to normal forces are plotted by squares, the ones of the regeneration due to sliding forces or rolling moments by triangles and circles, respectively.

Snapshots of the equilibrium configurations under uniaxial compression. The initial configuration (a) is a prepared sample in Sec. II A , whose packing fraction is . The configurations (b) and (c) show the beginning of the single-mode plastic regime ( ) and the multimode plastic regime ( ), respectively and (d) shows . The red circles express the correlation lengths; their diameters are , which will be evaluated in the later section (Sec. III F ).

Snapshots of the equilibrium configurations under uniaxial compression. The initial configuration (a) is a prepared sample in Sec. II A , whose packing fraction is . The configurations (b) and (c) show the beginning of the single-mode plastic regime ( ) and the multimode plastic regime ( ), respectively and (d) shows . The red circles express the correlation lengths; their diameters are , which will be evaluated in the later section (Sec. III F ).

The density-density correlation functions C(r) of compressed networks are shown. The solid line shows the initial configuration and the dashed or dotted lines the every second compression step (i = 1, 3, 5,…). They are averaged values over ten runs of simulations.

The density-density correlation functions C(r) of compressed networks are shown. The solid line shows the initial configuration and the dashed or dotted lines the every second compression step (i = 1, 3, 5,…). They are averaged values over ten runs of simulations.

(a) The fractal dimension profiles evaluated by Eq. (13) are plotted. The solid line shows the initial configuration , and the dashed or dotted solid lines the every second compression step (i = 1, 3, 5,…). The numerical values indicate the packing fractions. (b) The -dependence of the correlation length (see the definition in the text).

(a) The fractal dimension profiles evaluated by Eq. (13) are plotted. The solid line shows the initial configuration , and the dashed or dotted solid lines the every second compression step (i = 1, 3, 5,…). The numerical values indicate the packing fractions. (b) The -dependence of the correlation length (see the definition in the text).

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