^{1}, Charlotte Pellet

^{2}, Michel Cloitre

^{2}and Roger Bonnecaze

^{3,a)}

### Abstract

Oscillatory shear is a widely used characterization technique for complex fluids but the microstructural changes that produce the material response are not well understood. We apply a recent micromechanical model to soft particle glasses subjected to oscillatory flow. We use particle scale simulations at small, intermediate and large strain amplitudes to determine their microstructure, particle scale mobility, and macroscopic rheology. The macroscopic properties computed from simulations quantitatively agree with experimental measurements on well-characterized microgel suspensions, which validate the model. At the mesoscopic scale, the evolution of the particle pair distribution during a cycle reveals the physical mechanisms responsible for yielding and flow and also leads to quantitative prediction of shear stress. At the local scale, the particles remain trapped inside their surrounding cage below the yield strain and yielding is associated with the onset of large scale rearrangements and shear-induced diffusion. This multiscale analysis thus highlights the distinct microscopic events that make these glasses exhibit a combination of solid-like and liquid-like behavior.

The authors gratefully acknowledge support from the National Science Foundation (Grant No. 0854420) and the computational resources of the Texas Advanced Computing Center (TACC) at The University of Texas at Austin. M.C. thanks Professor Dimitris Vlassopoulos for enlightening discussions about LAOS. L.M. acknowledges support of an Eiffel Excellence scholarship from the French Ministry of Foreign Affairs.

I. INTRODUCTION

II. SIMULATIONS AND EXPERIMENTS

A. Simulation technique

B. Experiments

1. Microgel description and sample preparation

2. Rheological measurements

3. Material properties

III. RESULTS

A. Particle scale dynamics

1. Particle mean square displacements: Effect of strain amplitude

2. Mean square displacements: Effect of frequency

3. Shear-induced diffusivity

B. Microstructure of suspensions during oscillatory shear flow

1. Microstructure at rest

2. Microstructure at small amplitude oscillatory shear

3. Microstructure at medium amplitude oscillatory shear

4. Microstructure at large amplitude oscillatory shear

C. Macroscopic oscillatory shear rheology

1. Viscoelastic moduli at low strain amplitude

2. Effective viscoelastic moduli during large amplitude oscillations

3. Bowditch-Lissajous plots for arbitrary strain amplitudes

IV. DISCUSSION

A. Structural evolution within oscillatory cycles

B. From microstructure to macroscopic oscillatory shear rheology

V. CONCLUDING REMARKS

### Key Topics

- Suspensions
- 40.0
- Elasticity
- 29.0
- Elastic moduli
- 27.0
- Yield stress
- 22.0
- Microgels
- 17.0

## Figures

(a) Periodic simulation box. (b) Imposed oscillatory shear rate. (c) Schematic diagram showing pairwise interactions between particles where n // and n ⊥ are directions parallel and perpendicular to the particle–particle facet, respectively; u α and u β are the velocity of particles α and β, respectively; rα β is the center-to center distance between α and β; u α β ,// is the parallel component of the relative velocity between α and β. [(c) Has been reproduced from Seth et al. (2011) .]

(a) Periodic simulation box. (b) Imposed oscillatory shear rate. (c) Schematic diagram showing pairwise interactions between particles where n // and n ⊥ are directions parallel and perpendicular to the particle–particle facet, respectively; u α and u β are the velocity of particles α and β, respectively; rα β is the center-to center distance between α and β; u α β ,// is the parallel component of the relative velocity between α and β. [(c) Has been reproduced from Seth et al. (2011) .]

Theoretical flow curve (——) given by Eq. (4) and experimental flow curve (●) in the same set of coordinates for the microgel suspension at c = 2 wt. % with = 18 kPa (σ y = 35 Pa, γ y = 0.067, G 0 = 510 Pa, and η = mPa s). The inset shows the variations of G 0/ versus ϕ computed from simulations from which the effective volume fraction of the experimental suspension is determined.

Theoretical flow curve (——) given by Eq. (4) and experimental flow curve (●) in the same set of coordinates for the microgel suspension at c = 2 wt. % with = 18 kPa (σ y = 35 Pa, γ y = 0.067, G 0 = 510 Pa, and η = mPa s). The inset shows the variations of G 0/ versus ϕ computed from simulations from which the effective volume fraction of the experimental suspension is determined.

Mean square displacements of particles in the x-, y-, and z-directions versus oscillation number for different strain amplitudes: (a) γ 0 / γ y = 3.0, 1.5, 0.3 (top to bottom); (b) γ 0/γ y = 30, 15, 3.0 (top to bottom). The frequency is .

Mean square displacements of particles in the x-, y-, and z-directions versus oscillation number for different strain amplitudes: (a) γ 0 / γ y = 3.0, 1.5, 0.3 (top to bottom); (b) γ 0/γ y = 30, 15, 3.0 (top to bottom). The frequency is .

Mean square displacements of particles in the x-, y-, and z-directions at small and large strain amplitudes versus oscillation number for different frequencies. (a) Small strain amplitude (γ 0/γ y = 0.09) at frequencies of (top to bottom). (b) Large strain amplitude (γ 0/γ y = 30) at frequencies (top to bottom).

Mean square displacements of particles in the x-, y-, and z-directions at small and large strain amplitudes versus oscillation number for different frequencies. (a) Small strain amplitude (γ 0/γ y = 0.09) at frequencies of (top to bottom). (b) Large strain amplitude (γ 0/γ y = 30) at frequencies (top to bottom).

Shear-induced diffusion coefficients computed from the mean square displacements of particles. (a) Variations with the strain amplitude of the diffusion coefficients Di (i = x, y, z) computed from (◇), (△), and (▽), at nondimensional frequency ; the data for γ 0 / γ y < 1 have been estimated from the last computed oscillation where the mean square displacements approach their plateau values. (b) Variations with frequency of the nondimensional averaged diffusion coefficient, D = (D x + D y + D z)/3 at γ 0/γ y = 30 (◼) and γ 0/γ y = 3.0 (●). (c) Variations of the averaged diffusion coefficients D/D 0 for γ 0/γ y > 1 with the nondimensional shear-rate amplitude [squares represent averaged data from (a); same symbols as in (b)].

Shear-induced diffusion coefficients computed from the mean square displacements of particles. (a) Variations with the strain amplitude of the diffusion coefficients Di (i = x, y, z) computed from (◇), (△), and (▽), at nondimensional frequency ; the data for γ 0 / γ y < 1 have been estimated from the last computed oscillation where the mean square displacements approach their plateau values. (b) Variations with frequency of the nondimensional averaged diffusion coefficient, D = (D x + D y + D z)/3 at γ 0/γ y = 30 (◼) and γ 0/γ y = 3.0 (●). (c) Variations of the averaged diffusion coefficients D/D 0 for γ 0/γ y > 1 with the nondimensional shear-rate amplitude [squares represent averaged data from (a); same symbols as in (b)].

Microstructure of soft particle glass at rest; the volume fraction is ϕ = 0.80. (a) Static radial distribution function; (b) Pair distribution function shown in the x–y plane. (c) Pair distribution function shown in the azimuthal r–θ plane with the most probable center-to-center distance indicated by a white dashed-dotted line and a black arrow.

Microstructure of soft particle glass at rest; the volume fraction is ϕ = 0.80. (a) Static radial distribution function; (b) Pair distribution function shown in the x–y plane. (c) Pair distribution function shown in the azimuthal r–θ plane with the most probable center-to-center distance indicated by a white dashed-dotted line and a black arrow.

Microstructure of soft particle glass (ϕ = 0.80) subjected to small amplitude oscillations ( = 0.09; ηω/ = 2 × 10−8). (a) Variations of the strain (- - -) and stress (—) waveforms over one cycle and positions of the five characteristic points where g(r) is presented. (b) Pair distribution functions in the azimuthal r– plane; the most probable center-to-center separation at rest is indicated in the maximum and zero stress states by a white dashed-dotted line and a black arrow. (c) g 2,−2(r) spherical harmonics.

Microstructure of soft particle glass (ϕ = 0.80) subjected to small amplitude oscillations ( = 0.09; ηω/ = 2 × 10−8). (a) Variations of the strain (- - -) and stress (—) waveforms over one cycle and positions of the five characteristic points where g(r) is presented. (b) Pair distribution functions in the azimuthal r– plane; the most probable center-to-center separation at rest is indicated in the maximum and zero stress states by a white dashed-dotted line and a black arrow. (c) g 2,−2(r) spherical harmonics.

Microstructure of soft particle glass (ϕ = 0.80) subjected to medium amplitude oscillations ( = 3.0; ηω/E * = 2 × 10−8). (a) Variations of the strain (- - -) and stress (—) waveforms over one cycle and positions of the six characteristic points where g(r) is presented. (b) Pair distribution functions in the azimuthal r– plane; the most probable center-to-center separation at rest indicated in the zero stress states by a white dashed-dotted line and a black arrow. (c) g 2,−2(r) spherical harmonics.

Microstructure of soft particle glass (ϕ = 0.80) subjected to medium amplitude oscillations ( = 3.0; ηω/E * = 2 × 10−8). (a) Variations of the strain (- - -) and stress (—) waveforms over one cycle and positions of the six characteristic points where g(r) is presented. (b) Pair distribution functions in the azimuthal r– plane; the most probable center-to-center separation at rest indicated in the zero stress states by a white dashed-dotted line and a black arrow. (c) g 2,−2(r) spherical harmonics.

Microstructure of soft particle glass (ϕ = 0.80) subjected to large amplitude oscillations ( = 30; ηω/E * = 2 × 10−8). (a) Variations of the strain (- - -) and stress (—) waveforms over one cycle and positions of the six characteristic points where g(r) is presented. (b) Pair distribution functions in the azimuthal r– plane; the most probable center-to-center separation at rest is indicated in the zero stress states by a white dashed-dotted line and a black arrow; (c) g 2,−2(r) spherical harmonics.

Microstructure of soft particle glass (ϕ = 0.80) subjected to large amplitude oscillations ( = 30; ηω/E * = 2 × 10−8). (a) Variations of the strain (- - -) and stress (—) waveforms over one cycle and positions of the six characteristic points where g(r) is presented. (b) Pair distribution functions in the azimuthal r– plane; the most probable center-to-center separation at rest is indicated in the zero stress states by a white dashed-dotted line and a black arrow; (c) g 2,−2(r) spherical harmonics.

Storage modulus (◻ and —) and loss modulus (◇ and −−) versus reduced frequency from simulations (symbols) and experiments (lines) in the low strain amplitude or linear regime at γ 0/γy = 0.09. A reference slope of 0.5 is shown.

Storage modulus (◻ and —) and loss modulus (◇ and −−) versus reduced frequency from simulations (symbols) and experiments (lines) in the low strain amplitude or linear regime at γ 0/γy = 0.09. A reference slope of 0.5 is shown.

Storage modulus (◻ and —), loss modulus (◇ and ----), and stress amplitude σ 0 (● and ……) as functions of strain amplitude γ 0/γ y, from simulations (symbols) and experiments (lines) at a frequency of ηω/Ε∗ = 2 × 10−8 rad/s. Dotted lines represent power law variations with exponents μ and ν, respectively, as discussed in the text.

Storage modulus (◻ and —), loss modulus (◇ and ----), and stress amplitude σ 0 (● and ……) as functions of strain amplitude γ 0/γ y, from simulations (symbols) and experiments (lines) at a frequency of ηω/Ε∗ = 2 × 10−8 rad/s. Dotted lines represent power law variations with exponents μ and ν, respectively, as discussed in the text.

Bowditch–Lissajous plots from simulations and experiments at different strain amplitudes. Left to right: linear viscoelastic regime [γ 0/γ y = 0.09; panels (a) and (d)]; medium amplitude regime [inner to outer: γ 0/γ y = 0.09, 1.5, 3.0; panels (b) and (e)]; large amplitude regime [inner to outer: γ 0/γ y = 3.0, 15, 30, 60; panels (c) and (f)]. The symbols in panels (a)–(c) represent the shear stress values which are predicted from the g 2,−2(r) spherical harmonics as discussed in the text.

Bowditch–Lissajous plots from simulations and experiments at different strain amplitudes. Left to right: linear viscoelastic regime [γ 0/γ y = 0.09; panels (a) and (d)]; medium amplitude regime [inner to outer: γ 0/γ y = 0.09, 1.5, 3.0; panels (b) and (e)]; large amplitude regime [inner to outer: γ 0/γ y = 3.0, 15, 30, 60; panels (c) and (f)]. The symbols in panels (a)–(c) represent the shear stress values which are predicted from the g 2,−2(r) spherical harmonics as discussed in the text.

Cage modulus versus strain amplitude at ηω/E * = 2 × 10−8 from simulations (○) and experiments (●). For comparison, the values of the low frequency storage modulus at low strain amplitude are also plotted (◻: simulations; —: experiments).

Cage modulus versus strain amplitude at ηω/E * = 2 × 10−8 from simulations (○) and experiments (●). For comparison, the values of the low frequency storage modulus at low strain amplitude are also plotted (◻: simulations; —: experiments).

Flowing portions of the BL plots for different strain amplitudes and frequencies (symbols) collapsed and superimposed to the flow curve from steady shear (lines). For the sake of comparison between experiments and simulations, the data are represented in the set of reduced coordinates exemplified in Eq. (4) . Simulations (a) and experiments (b).

Flowing portions of the BL plots for different strain amplitudes and frequencies (symbols) collapsed and superimposed to the flow curve from steady shear (lines). For the sake of comparison between experiments and simulations, the data are represented in the set of reduced coordinates exemplified in Eq. (4) . Simulations (a) and experiments (b).

## Tables

Material properties of the experimental and simulated suspensions with c = 2 wt. % and ϕ = 0.80, respectively. The storage modulus and yield stress of the experimental microgel suspensions are G 0 = 510 Pa and σ y = 35 Pa.

Material properties of the experimental and simulated suspensions with c = 2 wt. % and ϕ = 0.80, respectively. The storage modulus and yield stress of the experimental microgel suspensions are G 0 = 510 Pa and σ y = 35 Pa.

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