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PHYSICAL REVIEW LETTERS
PRL 111, 268301 (2013)
Microscopic Origin of Internal Stresses in Jammed Soft Particle Suspensions *
Lavanya Mohan,1 Roger T. Bonnecaze,1,* and Michel Cloitre2,†
Department of Chemical Engineering and Texas Materials Institute, The University of Texas at Austin, Austin, Texas 78712, USA † Matière Molle et Chimie (UMR 7167, ESPCI-CNRS), ESPCI ParisTech 10 rue Vauquelin, 75005 Paris, France (Received 14 August 2013; published 26 December 2013) The long time persistence of mechanical stresses is a generic property of glassy materials. Here we identify the microscopic mechanisms that control internal stresses in highly concentrated suspensions of soft particles brought to rest from steady flow. The persistence of the asymmetric angular distortions which characterize the pair distribution function during flow is at the origin of the internal stresses. Their long time evolution is driven by in-cage rearrangements of the elastic contacts between particles. The trapped macroscopic stress is related to the solvent viscosity, particle elasticity and volume fraction through a universal scaling derived from simulations and experiments. DOI: 10.1103/PhysRevLett.111.268301
PACS numbers: 82.70.-y, 61.43.Fs, 83.60.-a, 83.85.St
Many amorphous materials with countless applications are processed in the liquid state, where they are malleable, and quenched to the solid state. Solidification can be achieved by cooling in thermoplastic polymers and metals, interruption of flow when shaping ceramic pastes and latex coatings, or chemical reactions in gels or thermosets. Upon solidification the structure may not instantaneously equilibrate and some of the stress existing in the liquid state remains trapped for long periods of time, giving rise to an internal or residual stress. The importance of internal stresses has been long recognized in materials like polymers [1], inorganic glasses [2], and metallic glasses [3]. Internal stresses can adversely affect product performance but, when cleverly controlled, they can be used to create advanced materials with improved properties. In soft matter, the relaxation of internal stress upon flow cessation has been invoked to explain the spontaneous slow dynamics observed in a variety of glassy [4–11] and biological materials [12,13]. A recent investigation has identified the volume fraction and the shear rate during initial flow as the key parameters that control internal stress in hard sphere glasses [14]. However, important questions concerning the generality of this description and the connection between internal stress and particle scale mechanisms remain open. In this Letter we identify the microscopic origin of internal stresses in suspensions of soft and deformable particles above close packing where they are in contact and jammed and interact via soft elastic repulsions, namely, soft particle glasses. This class of materials is singularly distinct from hard sphere glasses where excluded volume interactions and entropic effects exclusively dominate [15,16]. Examples for industrial applications include microgel pastes, concentrated emulsions, and multilamellar vesicles [17]. Our approach combines particle scale simulations and well-defined rheological experiments on microgel suspensions where the constituent properties are systematically varied. We show that the stress exhibits 0031-9007=13=111(26)=268301(5)
a two step relaxation after flow cessation: a rapid drop at short time where the particles move ballistically, which is followed by an extremely slow decay where the particles are trapped but rearrange their positions locally. The internal stress is due to the persistence in the arrested state of the asymmetric angular distortions which characterize the pair distribution function during flow. The rapid initial decay is driven by the unbalanced contact forces resulting from the preshear flow, which also controls the amplitude of the short time relaxation. This is reflected in a scaling expression which relates the internal stress to the preshear stress and quantitatively predicts the internal stress of any jammed soft particle suspension from a limited number of material properties, namely, the solvent viscosity, the particle elasticity, and the volume fraction. The experimental system is a dense suspension of polyelectrolyte microgel particles which are swollen with water or water-glycerol mixtures of viscosity ηS, each with a radius R ≈ 150 nm. The synthesis and the preparation of the microgels have been described elsewhere [18]. The volume fraction ϕ can be readily varied by increasing the polymer concentration. Experiments reported here are performed in the jammed regime where particles are closely packed and interact elastically through their contacting facets. We use two batches of microgels with different contact modulus E , which depends on the crosslink density of the particles (E ¼ E=2 ð1 − ν2 Þ; E: Young's modulus; ν ¼ 0.5: Poisson's ratio for incompressible spheres). E and ϕ are determined using a procedure previously developed [19]. Rheological measurements are carried out using an Anton Paar MCR 501 rheometer mounted with a cone and Peltier plate geometry with a diameter of 50 mm, a 2° angle, and a truncation of 48 μm. The shearing surfaces are sandblasted to provide a surface roughness of 2–4 μm. In the experimental conditions used here, microgel suspensions flow homogeneously without any kind of strain localization including slip, shear banding, or fracture [20].
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A solvent trap is used to minimize water evaporation. All experiments are performed at 20.0 0.1 ∘ C. The yield stress σ y of each sample is first determined using different well established protocols—creep measurements, oscillatory strain sweeps, and steady shear experiments—all determinations agreeing within the experimental accuracy. Relaxation experiments are performed by preshearing the sample at a constant stress σ P > σ y for 30 s, setting the shear rate to zero at the end of preshear, and recording the shear stress. Figure 1(a) presents the typical variations of the shear stress σðtÞ for different preshear conditions. The stress is constant during preshear and drops rapidly to a slowly decaying plateau on flow cessation. The internal stress σ I is determined by linear extrapolation of the stress measured over a short time interval (< 50 s) to the instant of flow cessation. The internal stress is larger for smaller preshear stress and becomes quite significant for preshear stresses approaching the yield stress. Figure 1(b) shows stress relaxation data for soft glasses with varying constituent properties (ϕ, ηS , and E ) for similar preshear stresses. The internal stress sensitively depends on ϕ and E , increasing either results in a significantly larger internal stress on flow cessation, while changing ηS has little effect. Also note that the steady state stress for a given shear rate will depend on all the above parameters.
FIG. 1 (color online). Stress relaxation on flow cessation (a) Experiments with microgels (ϕ¼0.82, ηs ¼ 14 m Pa · s, E ≅ 18 kPa, σ y ¼ 39 Pa) for different preshear stresses; (σ P , γ_ ) are (60 Pa, 0.3 s−1 ), (100 Pa, 4.0 s−1 ), (153 Pa, 14.4 s−1 ), (250 Pa, 53.2 s−1 ), and (443 Pa, 200 s−1 ), respectively; the inset shows the long term behavior for σ P =σ y ¼ 3.8. (b) Rapid initial relaxation on flow cessation from a stress of 150–170 Pa in samples with varying constituent properties. (c) Simulations (φ ¼ 0.80) with steady preshear rates γ_ τ0 ¼ 10−8 , 10−7 , 10−6 , 10−5 , and 10−4 in increasing order. (d) Scaled experimental results; for better comparison of simulations and experiments the instant of flow cessation has been shifted.
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In order to elucidate the microstructural origin of the internal stress, we perform numerical simulations based on a micromechanical framework that has been successfully used to describe jammed suspensions of soft particles in steady [16] and oscillatory shear flow [19]. Soft glasses are modeled as three-dimensional random packings of 104 non-Brownian elastic spheres (polydispersity: 10%; elastic modulus: E ) confined in a cubic box, which are dispersed in a solvent (viscosity: ηS ) at large volume fractions (ϕ ¼ 0.8). The particles experience pairwise interactions of two types: repulsive elastic forces and solvent mediated elastohydrodynamic (EHD) forces [16, 17]. The packings are subjected to constant shear rates for five unit strains allowing steady state to be achieved, and then allowed to relax at zero shear rates. Figure 1(c) shows results of simulations for five preshearing conditions. Each curve is an average over five different initial configurations. Stress is scaled by the yield stress and time by the microscopic time τ0 ¼ ηs =E , which represents the ratio of viscous to elastic forces associated with EHD lubrication [15, 16]. For the parameters relevant to our experiments (ηs ≍10−3 Pa · s; E ≈ 104 Pa), τ0 ≈ 10−7 s. The simulations capture the initial preshear, the rapid short time stress relaxation, and the beginnings of the long time relaxation. The computational time required to simulate out to the longest times accessible in experiments is prohibitive. The flow properties during preshear have been studied in detail in a previous paper [16]; in this regime, the stress is dominated by the elastic contact forces associated with the alteration of the microstructure [16, 21]. Upon flow cessation, which is the subject of this Letter, the stress drops and the closer the preshear stress is to the yield stress, the lesser the stress relaxes. In the following, the internal stress from the simulation is taken to be the value of the stress at the last instant computed. The reasonable agreement between the results shown in Fig. 1(c) and the experimental data replotted using the same set of reduced variables in Fig. 1(d) shows that the simulations successfully capture the important trends found in experiments. Simulations reveal the particle-scale mechanisms through which soft glasses relax the stress. Figure 2(a) shows the time evolution of the average number of contacts per particle, N. While N is much smaller than at equilibrium during flow, it quickly returns to its equilibrium value (≅ 9.35) for all preshear stresses. Figure 2(b) shows the time variation of the particle mean square displacements after preshear (t > t0 ). First, the mean square displacements are quadratic in time (Δr2 ¼ V 2 t2 ), identifying ballistic motion as the particle scale mechanism associated with the initial fast relaxation that takes place on flow cessation. The characteristic velocity of ballistic motion V is the largest for the largest preshear, where stress relaxation is the fastest and the ultimate trapped internal stress the smallest. Second, all the curves tend to a nearly constant plateau value which is less than one particle radius for all preshearing stresses, indicating that the particles
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FIG. 2 (color online). (a) Mean number of contacts per particle versus time for different preshear rates (from bottom to top: γ_ τ0 ¼ 10−4 , 10−5 , 10−6 , 10−7 , 10−8 ). (b) Mean square displacements (same rates, but from top to bottom). (c) Time evolution of the pair distribution function in polar coordinates in the flowgradient plane (_γ τ0 ¼ 10−4 ). The dashed white lines indicate the equilibrium mean center-to-center distance.
are quickly trapped in cages and that rearrangements associated with subsequent relaxation are local contact rearrangements. Figure 2(c) shows the evolution of the pair distribution functions computed in the flow-velocity gradient plane in polar coordinates (r, θ), where θ is measured from the positive flow direction [19] at three different times: during preshear (t0 ), at intermediate time during relaxation (t1 ), and at the beginning of the internal stress plateau (t2 ). For comparison, we also present the mean particle center-to-center separation in a fully equilibrated packing [22]. During preshear (t ¼ t0 ), we observe a clear angular distortion of the microstructure revealing accumulation of neighboring particles in the compressive upstream quadrant (π=2 < θ < π) and depletion in the extensional quadrant (0 < θ < π=2). Most particles are more compressed than at equilibrium. The asymmetry of the pair distribution function and the increased compression of particles persist during the short time relaxation (t ¼ t1 ), albeit reduced. At the end of the simulation (t ¼ t2 ), the mean center-to-center distance has relaxed to its equilibrated value but some degree of angular
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asymmetry seems to persist. Thus, we have identified the two distinct processes that constitute the microstructural signature of stress relaxation after flow cessation. To link the evolution of the pair distribution function which characterizes the microstructure and the shear stress associated with macroscopic rheology, we now quantitatively analyze the angular distortion of the microstructure by expanding the three-dimensional pair distribution function into an orthogonal series of harmonic in spherical P functions Pl coordinates: gðrÞ ¼ gðrÞ þ ∞ l¼1 m¼−l glm ðrÞY lm ðθ; ϕÞ [19, 23]. The angular asymmetry due to the accumulationdepletion of particles between the compression and extension directions is quantified by the coefficient g2;−2 ðrÞ [16, 23]. It constitutes the most important contributor to the stress: pffiffiffiffiffiffiffiffiffiffi R 3 σ ¼ −ð3ϕ=4πR3 Þ2 π=15 2R where r¼0 r fðrÞg2;−2 ðrÞdr fðrÞ is the magnitude of the elastic force between two particles [24]. Figures 3(a) and 3(b) show the coefficients g2;−2 ðrÞ computed for different preshear rates during preshear (t < t0 ) and at the beginning of the stress plateau (t ¼ t2 ). The asymmetry in the pair distribution function due to the accumulation-depletion mechanism appears clearly at t < t0 and persists at t ¼ t2 . A deeper minimum of g2;−2 ðrÞ indicates larger angular distortion of microstructure, yielding a higher stress. Interestingly, the variations of the depth of minima inverts from Fig. 3(a) to 3(b). In Fig. 3(a), the largest preshear rate and stress is associated with the largest asymmetry, i.e., the largest shear stress, resulting in Fig. 3(b) with the smallest asymmetry, i.e., the smallest internal stress. Also, note that the position of the minima in Fig. 3(b) does not depend on the preshear stress expressing that the compression of particles quickly relax to the equilibrium value. These results reveal the following microscopic picture of internal stress build up and relaxation in soft glasses. During preshear the macroscopic flow distorts the microstructure and causes an asymmetry in the pair distribution function, reflecting the existence of unbalanced elastic contact forces. On flow cessation, the particle distribution prevailing during preshear is instantaneously frozen and the contact forces drive the material back to equilibrium according to two different processes. First, the centerto-center distance and the average number of contacts relax
FIG. 3 (color online). Spherical harmonic coefficient g2;−2 ðrÞ: (a) during preshear at different rates (from left to right: γ_ τ0 ¼ 10−4 , 10−5 , 10-6 , 10−7 , 10−8 ); (b) at the final time of relaxation in simulations (same rates, from top to bottom).
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through ballistic motion to their equilibrium values [Figs. 2(a) and 2(b)]. This process is fast because only central forces at particle-particle contacts are important in order to change particle compression. However, the isotropy of gðrÞ is not fully restored and some asymmetry in the angular distribution of particles persists. It is this metastable asymmetry in the pair distribution function that is the origin of the internal stress. The larger unbalanced contact forces are, i.e., the driving forces—the more easily the isotropy of the pair distribution function is restored and the smaller is the internal stress. The subsequent relaxation of the internal stress is slow because the particles are trapped in cages so that further changes of the local topology between two contacting particles require collective rearrangements over long distances. In this description, the unbalanced contact forces created during the preshear flow drives the short time relaxation and controls the value of the internal stress after flow. Since flow induced rearrangements responsible for the pair distribution asymmetry occur only beyond the yield point [19], the internal stress must exclusively depend on the excess stress (σ P − σ y ). To support this prediction, we perform systematic experiments for different preshearing conditions and samples where the relevant material parameters (ηs ; ϕ; E ) are varied, and we measure the resulting internal stresses. To account for different values of the yield stress, we normalize the stresses by the yield stress and plot σ I =σ y as a function of ðσ P − σ y Þ=σ y . In Fig. 4 all the data collapse and form a line in log-linear coordinates showing that the internal stress varies logarithmically with the preshear stress. The raw data are presented in the inset to further
FIG. 4. Universal scaling for the effect of preshear stress σ P on internal stress σ I . Open symbols represent experimental data, varying the volume fraction: φ¼ 0.82 (open square), 0.94 (open circle) with ηs ¼ 1 mPa · s and E ¼ 18 kPa; the solvent viscosity ηs ¼ 5 (open upward triangle), 14 (open downward triangle), and 40 mPa · s (open left side facing triangle) with ϕ ≅ 0.79 and E ¼ 18 kPa; the particle softness: E ¼ 28 kPa with ϕ ¼ 0.73 (open diamond), 0.84 (crosses), and 0.88 (asterisk) and ηs ¼ 1 mPa · s. Black stars refer to simulation data. The inset shows the unscaled experimental data.
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support the efficiency of the scaling. It is interesting to note that, from the constitutive equation derived earlier [16], the reduced excess stress can be expressed as ðσ P -σ y Þ=σ y ∝ ð_γ ηS =γ 2y E Þ1=2 , where γ y represents the yield strain of the material which is a function of the volume fraction only [16]. Thus, this scaling shown in Fig. 4 also relates the material properties (ηs , E , ϕ) to the internal stress measured immediately after flow cessation. In Fig. 4, the simulation results also collapse reasonably well with the experimental data. We observe some deviation from the master curve at the largest preshear stress, which may be due to the finite size of the box which causes faster relaxation when the drive towards relaxation is larger. Additional larger scale simulations will be useful to explore this regime. In summary, we have elucidated the microstructural origin of internal stresses in jammed suspensions of soft particles, which are quenched from a sheared liquid state to an arrested solid state. The stress relaxes through two distinct processes: a rapid relaxation, where the radial distribution of contacts returns to equilibrium and the angular distribution partially relaxes; and then a slow relaxation where the angular distortions of the microstructure vanish. During the rapid relaxation, particles undergo ballistic motion over short distances before they stop and remain trapped in their local environment. This scenario distinguishes jammed suspensions of soft particles from thermal glasses near the glass transition where subdiffusive motion is observed [14]. Experimentally, ballistic motion has been found in the relaxation of depletion gels [10] and multilamellar vesicle suspensions [17]. The internal stress at the end of the rapid process is lower when the preshear stress or rate is higher. A similar trend has been observed in other glassy materials such as hard sphere thermal glasses [14] and laponite suspensions [11]. Our result for jammed soft particle suspensions suggests that this is a hallmark of glassy behavior. We have proposed a scaling, which goes beyond this qualitative trend, and allows one to predict quantitatively the internal stress for various processing conditions and material properties. Three parameters are important: at the microscopic level they are the solvent viscosity and the particle elasticity, which can be tuned by the nature and the architecture of the particles, and at the macroscopic level the concentration of the suspension. These predictions will apply to a large class of glassy systems involving soft spheres like emulsions, vesicles, and charged colloids. It will also be interesting to apply the same framework to investigate ultrasoft particles like star polymers, where thermal effects come into play and potentially provide new relaxation mechanisms [25]. Another question under current investigation is the connection between internal stress and slow relaxation and aging phenomena in soft particle glasses. The authors gratefully acknowledge support from the National Science Foundation (Grant –No. 0854420) and
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the computational resources of the Texas Advanced Computing Center (TACC) at The University of Texas at Austin. The experimental work was performed when L. M. was visiting the "Matière Molle et Chimie" laboratory at ESPCI ParisTech with the support of an Eiffel Excellence scholarship from the French Ministry of Foreign Affairs. L. M. thanks Charlotte Pellet for her support in performing the experiments.
*[email protected] *[email protected] [1] L. C. E. Struik, Internal Stresses, Dimensional Instabilities and Molecular Orientations in Plastics (Wiley, Brisbane, 1990). [2] D. J. Green, R. Tandon, and V. M. Sglavo, Science 283, 1295 (1999). [3] Y. Zhang, W. H. Wang, and A. L. Greer, Nat. Mater. 5, 857 (2006). [4] L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, Phys. Rev. Lett. 84, 2275 (2000). [5] M. Cloitre, R. Borrega, and L. Leibler, Phys. Rev. Lett. 85, 4819 (2000). [6] L. Ramos and L. Cipelletti, Phys. Rev. Lett. 87, 245503 (2001). [7] L. Ramos and L. Cipelletti, Phys. Rev. Lett. 94, 158301 (2005). [8] A. S. Negi and C. O. Osuji, Phys. Rev. E 80, 010404 (2009). [9] S. Q. Wang, S. Ravindranath, P. Boukany, M. Olechnowicz, R. P. Quirk, A. Halasa, and J. Mays, Phys. Rev. Lett. 97, 187801 (2006). [10] B. Chung, S. Ramakrishnan, R. Bandyopadhyay, D. Liang, C. F. Zukoski, J. L. Harden, and R. L. Leheny, Phys. Rev. Lett. 96, 228301 (2006).
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[11] A. S. Negi and C. O. Osuji, J. Rheol. 54, 943 (2010). [12] R. Bandyopadhyay, D. Liang, J. L. Harden, and R. L. Leheny, Solid State Commun. 139, 589 (2006). [13] O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti, and A. R. Bausch, Nat. Mater. 10, 236 (2011). [14] M. Ballauff, J. M. Brader, S. U. Egelhaaf, M. Fuchs, J. Horbach, N. Koumakis, M. Krüger, M. Laurati, K. J. Mutch, G. Petekidis, M. Siebenburger, Th. Voigtmann, and J. Zausch, Phys. Rev. Lett. 110, 215701 (2013). [15] M. Cloitre, R. Borrega, F. Monti, and L. Leibler, Phys. Rev. Lett. 90, 068303 (2003). [16] J. R. Seth, L. Mohan, C. Locatelli-Champagne, M. Cloitre, and R. T. Bonnecaze, Nat. Mater. 10, 838 (2011). [17] R. T. Bonnecaze and M. Cloitre, Adv. Polym. Sci. 236, 117 (2010). [18] M. Cloitre, R. Borrega, F. Monti, and L. Leibler, C.R. Physique 4, 221 (2003). [19] L. Mohan, C. Pellet, M. Cloitre, and R. T. Bonnecaze, J. Rheol. 57, 1023 (2013). [20] J. R. Seth, C. Locatelli-Champagne, F. Monti, R. T. Bonnecaze, and M. Cloitre, Soft Matter 8, 140 (2012). [21] B. P. Tighe, E. Woldhuis, J. J. C. Remmers, W. van Saarloos, and M. van Hecke, Phys. Rev. Lett. 105, 088303 (2010). [22] J. R. Seth, M. Cloitre, and R. T. Bonnecaze, J. Rheol. 50, 353 (2006). [23] J. F. Morris and B. Katyal, Phys. Fluids 14, 1920 (2002). [24] During flow, the dissipative stress is the sum of a contribution due to the elastic deformation of the particles and a hydrodynamic contribution due to viscous friction in the interstitial fluid between contacts. Following [16], the former contribution dominates, which allows to compute the stress from the pairwise elastic contact force fðrÞ. [25] B. M. Erwin, D. Vlassopoulos, M. Gauthier, and M. Cloitre, Phys. Rev. E 83, 061402 (2011).
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PRL 111, 268301 (2013)
Microscopic Origin of Internal Stresses in Jammed Soft Particle Suspensions *
Lavanya Mohan,1 Roger T. Bonnecaze,1,* and Michel Cloitre2,†
Department of Chemical Engineering and Texas Materials Institute, The University of Texas at Austin, Austin, Texas 78712, USA † Matière Molle et Chimie (UMR 7167, ESPCI-CNRS), ESPCI ParisTech 10 rue Vauquelin, 75005 Paris, France (Received 14 August 2013; published 26 December 2013) The long time persistence of mechanical stresses is a generic property of glassy materials. Here we identify the microscopic mechanisms that control internal stresses in highly concentrated suspensions of soft particles brought to rest from steady flow. The persistence of the asymmetric angular distortions which characterize the pair distribution function during flow is at the origin of the internal stresses. Their long time evolution is driven by in-cage rearrangements of the elastic contacts between particles. The trapped macroscopic stress is related to the solvent viscosity, particle elasticity and volume fraction through a universal scaling derived from simulations and experiments. DOI: 10.1103/PhysRevLett.111.268301
PACS numbers: 82.70.-y, 61.43.Fs, 83.60.-a, 83.85.St
Many amorphous materials with countless applications are processed in the liquid state, where they are malleable, and quenched to the solid state. Solidification can be achieved by cooling in thermoplastic polymers and metals, interruption of flow when shaping ceramic pastes and latex coatings, or chemical reactions in gels or thermosets. Upon solidification the structure may not instantaneously equilibrate and some of the stress existing in the liquid state remains trapped for long periods of time, giving rise to an internal or residual stress. The importance of internal stresses has been long recognized in materials like polymers [1], inorganic glasses [2], and metallic glasses [3]. Internal stresses can adversely affect product performance but, when cleverly controlled, they can be used to create advanced materials with improved properties. In soft matter, the relaxation of internal stress upon flow cessation has been invoked to explain the spontaneous slow dynamics observed in a variety of glassy [4–11] and biological materials [12,13]. A recent investigation has identified the volume fraction and the shear rate during initial flow as the key parameters that control internal stress in hard sphere glasses [14]. However, important questions concerning the generality of this description and the connection between internal stress and particle scale mechanisms remain open. In this Letter we identify the microscopic origin of internal stresses in suspensions of soft and deformable particles above close packing where they are in contact and jammed and interact via soft elastic repulsions, namely, soft particle glasses. This class of materials is singularly distinct from hard sphere glasses where excluded volume interactions and entropic effects exclusively dominate [15,16]. Examples for industrial applications include microgel pastes, concentrated emulsions, and multilamellar vesicles [17]. Our approach combines particle scale simulations and well-defined rheological experiments on microgel suspensions where the constituent properties are systematically varied. We show that the stress exhibits 0031-9007=13=111(26)=268301(5)
a two step relaxation after flow cessation: a rapid drop at short time where the particles move ballistically, which is followed by an extremely slow decay where the particles are trapped but rearrange their positions locally. The internal stress is due to the persistence in the arrested state of the asymmetric angular distortions which characterize the pair distribution function during flow. The rapid initial decay is driven by the unbalanced contact forces resulting from the preshear flow, which also controls the amplitude of the short time relaxation. This is reflected in a scaling expression which relates the internal stress to the preshear stress and quantitatively predicts the internal stress of any jammed soft particle suspension from a limited number of material properties, namely, the solvent viscosity, the particle elasticity, and the volume fraction. The experimental system is a dense suspension of polyelectrolyte microgel particles which are swollen with water or water-glycerol mixtures of viscosity ηS, each with a radius R ≈ 150 nm. The synthesis and the preparation of the microgels have been described elsewhere [18]. The volume fraction ϕ can be readily varied by increasing the polymer concentration. Experiments reported here are performed in the jammed regime where particles are closely packed and interact elastically through their contacting facets. We use two batches of microgels with different contact modulus E , which depends on the crosslink density of the particles (E ¼ E=2 ð1 − ν2 Þ; E: Young's modulus; ν ¼ 0.5: Poisson's ratio for incompressible spheres). E and ϕ are determined using a procedure previously developed [19]. Rheological measurements are carried out using an Anton Paar MCR 501 rheometer mounted with a cone and Peltier plate geometry with a diameter of 50 mm, a 2° angle, and a truncation of 48 μm. The shearing surfaces are sandblasted to provide a surface roughness of 2–4 μm. In the experimental conditions used here, microgel suspensions flow homogeneously without any kind of strain localization including slip, shear banding, or fracture [20].
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A solvent trap is used to minimize water evaporation. All experiments are performed at 20.0 0.1 ∘ C. The yield stress σ y of each sample is first determined using different well established protocols—creep measurements, oscillatory strain sweeps, and steady shear experiments—all determinations agreeing within the experimental accuracy. Relaxation experiments are performed by preshearing the sample at a constant stress σ P > σ y for 30 s, setting the shear rate to zero at the end of preshear, and recording the shear stress. Figure 1(a) presents the typical variations of the shear stress σðtÞ for different preshear conditions. The stress is constant during preshear and drops rapidly to a slowly decaying plateau on flow cessation. The internal stress σ I is determined by linear extrapolation of the stress measured over a short time interval (< 50 s) to the instant of flow cessation. The internal stress is larger for smaller preshear stress and becomes quite significant for preshear stresses approaching the yield stress. Figure 1(b) shows stress relaxation data for soft glasses with varying constituent properties (ϕ, ηS , and E ) for similar preshear stresses. The internal stress sensitively depends on ϕ and E , increasing either results in a significantly larger internal stress on flow cessation, while changing ηS has little effect. Also note that the steady state stress for a given shear rate will depend on all the above parameters.
FIG. 1 (color online). Stress relaxation on flow cessation (a) Experiments with microgels (ϕ¼0.82, ηs ¼ 14 m Pa · s, E ≅ 18 kPa, σ y ¼ 39 Pa) for different preshear stresses; (σ P , γ_ ) are (60 Pa, 0.3 s−1 ), (100 Pa, 4.0 s−1 ), (153 Pa, 14.4 s−1 ), (250 Pa, 53.2 s−1 ), and (443 Pa, 200 s−1 ), respectively; the inset shows the long term behavior for σ P =σ y ¼ 3.8. (b) Rapid initial relaxation on flow cessation from a stress of 150–170 Pa in samples with varying constituent properties. (c) Simulations (φ ¼ 0.80) with steady preshear rates γ_ τ0 ¼ 10−8 , 10−7 , 10−6 , 10−5 , and 10−4 in increasing order. (d) Scaled experimental results; for better comparison of simulations and experiments the instant of flow cessation has been shifted.
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In order to elucidate the microstructural origin of the internal stress, we perform numerical simulations based on a micromechanical framework that has been successfully used to describe jammed suspensions of soft particles in steady [16] and oscillatory shear flow [19]. Soft glasses are modeled as three-dimensional random packings of 104 non-Brownian elastic spheres (polydispersity: 10%; elastic modulus: E ) confined in a cubic box, which are dispersed in a solvent (viscosity: ηS ) at large volume fractions (ϕ ¼ 0.8). The particles experience pairwise interactions of two types: repulsive elastic forces and solvent mediated elastohydrodynamic (EHD) forces [16, 17]. The packings are subjected to constant shear rates for five unit strains allowing steady state to be achieved, and then allowed to relax at zero shear rates. Figure 1(c) shows results of simulations for five preshearing conditions. Each curve is an average over five different initial configurations. Stress is scaled by the yield stress and time by the microscopic time τ0 ¼ ηs =E , which represents the ratio of viscous to elastic forces associated with EHD lubrication [15, 16]. For the parameters relevant to our experiments (ηs ≍10−3 Pa · s; E ≈ 104 Pa), τ0 ≈ 10−7 s. The simulations capture the initial preshear, the rapid short time stress relaxation, and the beginnings of the long time relaxation. The computational time required to simulate out to the longest times accessible in experiments is prohibitive. The flow properties during preshear have been studied in detail in a previous paper [16]; in this regime, the stress is dominated by the elastic contact forces associated with the alteration of the microstructure [16, 21]. Upon flow cessation, which is the subject of this Letter, the stress drops and the closer the preshear stress is to the yield stress, the lesser the stress relaxes. In the following, the internal stress from the simulation is taken to be the value of the stress at the last instant computed. The reasonable agreement between the results shown in Fig. 1(c) and the experimental data replotted using the same set of reduced variables in Fig. 1(d) shows that the simulations successfully capture the important trends found in experiments. Simulations reveal the particle-scale mechanisms through which soft glasses relax the stress. Figure 2(a) shows the time evolution of the average number of contacts per particle, N. While N is much smaller than at equilibrium during flow, it quickly returns to its equilibrium value (≅ 9.35) for all preshear stresses. Figure 2(b) shows the time variation of the particle mean square displacements after preshear (t > t0 ). First, the mean square displacements are quadratic in time (Δr2 ¼ V 2 t2 ), identifying ballistic motion as the particle scale mechanism associated with the initial fast relaxation that takes place on flow cessation. The characteristic velocity of ballistic motion V is the largest for the largest preshear, where stress relaxation is the fastest and the ultimate trapped internal stress the smallest. Second, all the curves tend to a nearly constant plateau value which is less than one particle radius for all preshearing stresses, indicating that the particles
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FIG. 2 (color online). (a) Mean number of contacts per particle versus time for different preshear rates (from bottom to top: γ_ τ0 ¼ 10−4 , 10−5 , 10−6 , 10−7 , 10−8 ). (b) Mean square displacements (same rates, but from top to bottom). (c) Time evolution of the pair distribution function in polar coordinates in the flowgradient plane (_γ τ0 ¼ 10−4 ). The dashed white lines indicate the equilibrium mean center-to-center distance.
are quickly trapped in cages and that rearrangements associated with subsequent relaxation are local contact rearrangements. Figure 2(c) shows the evolution of the pair distribution functions computed in the flow-velocity gradient plane in polar coordinates (r, θ), where θ is measured from the positive flow direction [19] at three different times: during preshear (t0 ), at intermediate time during relaxation (t1 ), and at the beginning of the internal stress plateau (t2 ). For comparison, we also present the mean particle center-to-center separation in a fully equilibrated packing [22]. During preshear (t ¼ t0 ), we observe a clear angular distortion of the microstructure revealing accumulation of neighboring particles in the compressive upstream quadrant (π=2 < θ < π) and depletion in the extensional quadrant (0 < θ < π=2). Most particles are more compressed than at equilibrium. The asymmetry of the pair distribution function and the increased compression of particles persist during the short time relaxation (t ¼ t1 ), albeit reduced. At the end of the simulation (t ¼ t2 ), the mean center-to-center distance has relaxed to its equilibrated value but some degree of angular
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asymmetry seems to persist. Thus, we have identified the two distinct processes that constitute the microstructural signature of stress relaxation after flow cessation. To link the evolution of the pair distribution function which characterizes the microstructure and the shear stress associated with macroscopic rheology, we now quantitatively analyze the angular distortion of the microstructure by expanding the three-dimensional pair distribution function into an orthogonal series of harmonic in spherical P functions Pl coordinates: gðrÞ ¼ gðrÞ þ ∞ l¼1 m¼−l glm ðrÞY lm ðθ; ϕÞ [19, 23]. The angular asymmetry due to the accumulationdepletion of particles between the compression and extension directions is quantified by the coefficient g2;−2 ðrÞ [16, 23]. It constitutes the most important contributor to the stress: pffiffiffiffiffiffiffiffiffiffi R 3 σ ¼ −ð3ϕ=4πR3 Þ2 π=15 2R where r¼0 r fðrÞg2;−2 ðrÞdr fðrÞ is the magnitude of the elastic force between two particles [24]. Figures 3(a) and 3(b) show the coefficients g2;−2 ðrÞ computed for different preshear rates during preshear (t < t0 ) and at the beginning of the stress plateau (t ¼ t2 ). The asymmetry in the pair distribution function due to the accumulation-depletion mechanism appears clearly at t < t0 and persists at t ¼ t2 . A deeper minimum of g2;−2 ðrÞ indicates larger angular distortion of microstructure, yielding a higher stress. Interestingly, the variations of the depth of minima inverts from Fig. 3(a) to 3(b). In Fig. 3(a), the largest preshear rate and stress is associated with the largest asymmetry, i.e., the largest shear stress, resulting in Fig. 3(b) with the smallest asymmetry, i.e., the smallest internal stress. Also, note that the position of the minima in Fig. 3(b) does not depend on the preshear stress expressing that the compression of particles quickly relax to the equilibrium value. These results reveal the following microscopic picture of internal stress build up and relaxation in soft glasses. During preshear the macroscopic flow distorts the microstructure and causes an asymmetry in the pair distribution function, reflecting the existence of unbalanced elastic contact forces. On flow cessation, the particle distribution prevailing during preshear is instantaneously frozen and the contact forces drive the material back to equilibrium according to two different processes. First, the centerto-center distance and the average number of contacts relax
FIG. 3 (color online). Spherical harmonic coefficient g2;−2 ðrÞ: (a) during preshear at different rates (from left to right: γ_ τ0 ¼ 10−4 , 10−5 , 10-6 , 10−7 , 10−8 ); (b) at the final time of relaxation in simulations (same rates, from top to bottom).
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through ballistic motion to their equilibrium values [Figs. 2(a) and 2(b)]. This process is fast because only central forces at particle-particle contacts are important in order to change particle compression. However, the isotropy of gðrÞ is not fully restored and some asymmetry in the angular distribution of particles persists. It is this metastable asymmetry in the pair distribution function that is the origin of the internal stress. The larger unbalanced contact forces are, i.e., the driving forces—the more easily the isotropy of the pair distribution function is restored and the smaller is the internal stress. The subsequent relaxation of the internal stress is slow because the particles are trapped in cages so that further changes of the local topology between two contacting particles require collective rearrangements over long distances. In this description, the unbalanced contact forces created during the preshear flow drives the short time relaxation and controls the value of the internal stress after flow. Since flow induced rearrangements responsible for the pair distribution asymmetry occur only beyond the yield point [19], the internal stress must exclusively depend on the excess stress (σ P − σ y ). To support this prediction, we perform systematic experiments for different preshearing conditions and samples where the relevant material parameters (ηs ; ϕ; E ) are varied, and we measure the resulting internal stresses. To account for different values of the yield stress, we normalize the stresses by the yield stress and plot σ I =σ y as a function of ðσ P − σ y Þ=σ y . In Fig. 4 all the data collapse and form a line in log-linear coordinates showing that the internal stress varies logarithmically with the preshear stress. The raw data are presented in the inset to further
FIG. 4. Universal scaling for the effect of preshear stress σ P on internal stress σ I . Open symbols represent experimental data, varying the volume fraction: φ¼ 0.82 (open square), 0.94 (open circle) with ηs ¼ 1 mPa · s and E ¼ 18 kPa; the solvent viscosity ηs ¼ 5 (open upward triangle), 14 (open downward triangle), and 40 mPa · s (open left side facing triangle) with ϕ ≅ 0.79 and E ¼ 18 kPa; the particle softness: E ¼ 28 kPa with ϕ ¼ 0.73 (open diamond), 0.84 (crosses), and 0.88 (asterisk) and ηs ¼ 1 mPa · s. Black stars refer to simulation data. The inset shows the unscaled experimental data.
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support the efficiency of the scaling. It is interesting to note that, from the constitutive equation derived earlier [16], the reduced excess stress can be expressed as ðσ P -σ y Þ=σ y ∝ ð_γ ηS =γ 2y E Þ1=2 , where γ y represents the yield strain of the material which is a function of the volume fraction only [16]. Thus, this scaling shown in Fig. 4 also relates the material properties (ηs , E , ϕ) to the internal stress measured immediately after flow cessation. In Fig. 4, the simulation results also collapse reasonably well with the experimental data. We observe some deviation from the master curve at the largest preshear stress, which may be due to the finite size of the box which causes faster relaxation when the drive towards relaxation is larger. Additional larger scale simulations will be useful to explore this regime. In summary, we have elucidated the microstructural origin of internal stresses in jammed suspensions of soft particles, which are quenched from a sheared liquid state to an arrested solid state. The stress relaxes through two distinct processes: a rapid relaxation, where the radial distribution of contacts returns to equilibrium and the angular distribution partially relaxes; and then a slow relaxation where the angular distortions of the microstructure vanish. During the rapid relaxation, particles undergo ballistic motion over short distances before they stop and remain trapped in their local environment. This scenario distinguishes jammed suspensions of soft particles from thermal glasses near the glass transition where subdiffusive motion is observed [14]. Experimentally, ballistic motion has been found in the relaxation of depletion gels [10] and multilamellar vesicle suspensions [17]. The internal stress at the end of the rapid process is lower when the preshear stress or rate is higher. A similar trend has been observed in other glassy materials such as hard sphere thermal glasses [14] and laponite suspensions [11]. Our result for jammed soft particle suspensions suggests that this is a hallmark of glassy behavior. We have proposed a scaling, which goes beyond this qualitative trend, and allows one to predict quantitatively the internal stress for various processing conditions and material properties. Three parameters are important: at the microscopic level they are the solvent viscosity and the particle elasticity, which can be tuned by the nature and the architecture of the particles, and at the macroscopic level the concentration of the suspension. These predictions will apply to a large class of glassy systems involving soft spheres like emulsions, vesicles, and charged colloids. It will also be interesting to apply the same framework to investigate ultrasoft particles like star polymers, where thermal effects come into play and potentially provide new relaxation mechanisms [25]. Another question under current investigation is the connection between internal stress and slow relaxation and aging phenomena in soft particle glasses. The authors gratefully acknowledge support from the National Science Foundation (Grant –No. 0854420) and
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the computational resources of the Texas Advanced Computing Center (TACC) at The University of Texas at Austin. The experimental work was performed when L. M. was visiting the "Matière Molle et Chimie" laboratory at ESPCI ParisTech with the support of an Eiffel Excellence scholarship from the French Ministry of Foreign Affairs. L. M. thanks Charlotte Pellet for her support in performing the experiments.
*[email protected] *[email protected] [1] L. C. E. Struik, Internal Stresses, Dimensional Instabilities and Molecular Orientations in Plastics (Wiley, Brisbane, 1990). [2] D. J. Green, R. Tandon, and V. M. Sglavo, Science 283, 1295 (1999). [3] Y. Zhang, W. H. Wang, and A. L. Greer, Nat. Mater. 5, 857 (2006). [4] L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, Phys. Rev. Lett. 84, 2275 (2000). [5] M. Cloitre, R. Borrega, and L. Leibler, Phys. Rev. Lett. 85, 4819 (2000). [6] L. Ramos and L. Cipelletti, Phys. Rev. Lett. 87, 245503 (2001). [7] L. Ramos and L. Cipelletti, Phys. Rev. Lett. 94, 158301 (2005). [8] A. S. Negi and C. O. Osuji, Phys. Rev. E 80, 010404 (2009). [9] S. Q. Wang, S. Ravindranath, P. Boukany, M. Olechnowicz, R. P. Quirk, A. Halasa, and J. Mays, Phys. Rev. Lett. 97, 187801 (2006). [10] B. Chung, S. Ramakrishnan, R. Bandyopadhyay, D. Liang, C. F. Zukoski, J. L. Harden, and R. L. Leheny, Phys. Rev. Lett. 96, 228301 (2006).
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[11] A. S. Negi and C. O. Osuji, J. Rheol. 54, 943 (2010). [12] R. Bandyopadhyay, D. Liang, J. L. Harden, and R. L. Leheny, Solid State Commun. 139, 589 (2006). [13] O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti, and A. R. Bausch, Nat. Mater. 10, 236 (2011). [14] M. Ballauff, J. M. Brader, S. U. Egelhaaf, M. Fuchs, J. Horbach, N. Koumakis, M. Krüger, M. Laurati, K. J. Mutch, G. Petekidis, M. Siebenburger, Th. Voigtmann, and J. Zausch, Phys. Rev. Lett. 110, 215701 (2013). [15] M. Cloitre, R. Borrega, F. Monti, and L. Leibler, Phys. Rev. Lett. 90, 068303 (2003). [16] J. R. Seth, L. Mohan, C. Locatelli-Champagne, M. Cloitre, and R. T. Bonnecaze, Nat. Mater. 10, 838 (2011). [17] R. T. Bonnecaze and M. Cloitre, Adv. Polym. Sci. 236, 117 (2010). [18] M. Cloitre, R. Borrega, F. Monti, and L. Leibler, C.R. Physique 4, 221 (2003). [19] L. Mohan, C. Pellet, M. Cloitre, and R. T. Bonnecaze, J. Rheol. 57, 1023 (2013). [20] J. R. Seth, C. Locatelli-Champagne, F. Monti, R. T. Bonnecaze, and M. Cloitre, Soft Matter 8, 140 (2012). [21] B. P. Tighe, E. Woldhuis, J. J. C. Remmers, W. van Saarloos, and M. van Hecke, Phys. Rev. Lett. 105, 088303 (2010). [22] J. R. Seth, M. Cloitre, and R. T. Bonnecaze, J. Rheol. 50, 353 (2006). [23] J. F. Morris and B. Katyal, Phys. Fluids 14, 1920 (2002). [24] During flow, the dissipative stress is the sum of a contribution due to the elastic deformation of the particles and a hydrodynamic contribution due to viscous friction in the interstitial fluid between contacts. Following [16], the former contribution dominates, which allows to compute the stress from the pairwise elastic contact force fðrÞ. [25] B. M. Erwin, D. Vlassopoulos, M. Gauthier, and M. Cloitre, Phys. Rev. E 83, 061402 (2011).
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