Yield stress materials in soft condensed matter

A comprehensive review is presented of the physical behavior of yield stress materials in soft condensed matter, which encompasses a broad range of materials from colloidal assemblies and gels to emulsions and non-Brownian suspensions. All these disordered materials display a nonlinear flow behavior in response to external mechanical forces due to the existence of a finite force threshold for flow to occur: the yield stress. Both the physical origin and rheological consequences associated with this nonlinear behavior are discussed and an overview is given of experimental techniques available to measure the yield stress. Recent progress is discussed concerning a microscopic theoretical description of the flow dynamics of yield stress materials, emphasizing, in particular, the role played by relaxation time scales, the interplay between shear flow and aging behavior, the existence of inhomogeneous shear flows and shear bands, wall slip, and nonlocal effects in confined geometries.

Figure 1

Which fluid is more viscous: whipped cream or thick maple syrup? Slowly stirring both materials with a spoon suggests that syrup is more viscous, while observing the flattening of piles of each material with time suggests the opposite. In fact, the question is ill posed. The flow properties of whipped cream cannot be reduced to a single viscosity value because it is a yield stress material, whereas syrup is simply a very viscous fluid.

Figure 2

(a) The Bingham model (solid line) provides a reasonable fit to the experimental flow curve of whipped cream (crosses), with yield stress ${\sigma }_y\approx 33\mathrm{Pa}$. (b) The flow curves of syrup (plusses) and whipped cream (crosses) at low shear rates. Data points are connected by lines as guides to the eye. For stresses above 33 Pa, whipped cream flows more easily than syrup, while the opposite is true below 33 Pa. Using this enlarged scale, one can see that the flow curve of whipped cream below the yield stress is in fact not well described by the Bingham model [Eq. (1)], which simply predicts zero shear rate all the way up to ${\sigma }_y$.

Figure 3

(a) Soft glassy rheology. Evolution of the flow curves across the thermal colloidal glass transition for a suspension of PMMA hard spheres of size $a\approx 200\mathrm{nm}$. From [333]. (b) Jamming rheology. Evolution of the flow curves for an oil-in-water emulsion with droplet size $a\approx 3.2\mu \mathrm{m}$ across the athermal jamming transition. From [322]. In both cases, a yield stress appears above a certain critical density ${\varphi }_c$ (dashed line), which corresponds to the glass transition ${\varphi }_G$ for thermal systems in (a), and to the jamming transition ${\varphi }_J$ for athermal particles in (b). Although the emergence of solid behavior in both cases is conceptually very different, the flow curves of both materials are surprisingly similar.

Figure 4

Three-dimensional “jamming phase diagram” showing the reconstructed yield stress surface from numerical simulations as a function of the thermodynamic parameters temperature and density in a dimensionless representation (particle softness $k_{B}T/\epsilon$, volume fraction $\phi$, and stress $\sigma a^3/\epsilon$) for a model of soft harmonic particles. The thick lines represent the location of typical experimental measurements in various materials: foams (rightmost line) are mainly sensitive to jamming physics; PMMA hard spheres (black line) to glass physics; emulsions display an interesting interplay between glass and jamming transitions; poly($N$-isopropylacrylamide) (p-NIPAM) microgels (leftmost line) undergo a colloidal glass transition far from the jamming limit with no particular signature across the jamming density. From [197].

Figure 5

Viscosity vs shear stress in Carbopol. (a) From [20]. (b) A subsequent study of the same system. From [285]. The latter showed that the values of the low-stress viscosity plateau increase with measurement time (from 10 to 3000 s, colored symbols). The insets show that the plateau value increases as a power law with time, with exponent of about 0.6, indicating that the measured viscosities do not correspond to steady-state shear flows for shear stresses below the yield stress.

Figure 6

(a) The behavior of 0.1% wt Carbopol microgel under increasing and decreasing shear stresses shows that this material is nonthixotropic (filled circles, up; open circles, down). (b) Thixotropy of a 10% wt bentonite solution under an increasing and then decreasing stress ramp. From [286].

Figure 7

(a) A colloidal gel at rest, with a percolated structure and a yield stress of 5 Pa, and (b) just after flow, with individual flocs and no measurable yield stress. The gel is made up of $1.3\mu \mathrm{m}$ fluorescent PMMA particles and $3\times {10}^7\mathrm{Mw}$ polystyrene in a mixture of decalin and cyclohexyl bromide. From [44].

Figure 8

Changes in the local shear stresses (as indicated by the color coding) during a localized plastic event (top); the color coding gives the amplitude of the stress changes, and associated displacement field (bottom) observed in the numerical simulation of a quasistatically sheared model of atomic glass. Adapted from [406].

Figure 9

Flow curves predicted for a range of temperatures $T$ across the mode-coupling critical temperature $T_c$; $\epsilon =T_c-T$ is the distance to the critical temperature, and the shear rate is rescaled by a microscopic time unit $\tau$ to form a Péclet number $\mathrm{P}{\mathrm{e}}_0=\dot{\gamma }\tau$. The inset shows the discontinuous emergence of the yield stress at $T_c$. From [155].

Figure 10

Master curves showing a good collapse of the flow curves onto two branches, one for samples with $\varphi <{\varphi }_J$ and one for $\varphi >{\varphi }_J$, when stress and shear rate are rescaled with the distance to jamming to a certain power. The lines are supercritical and subcritical branches representing empirical fits of the master curve, respectively. The inset shows a fit of the low-shear viscosity to a power-law divergence. Flow curves were obtained for emulsions prepared with different volume fractions of the dispersed phase. From [322].

Figure 11

Various methods to determine a yield stress experimentally. (a) Extrapolation of the flow curve in the limit of vanishing shear rates. Experiments performed on a Carbopol microgel using roughened cone-and-plate fixtures. The black line is the best Herschel-Bulkley fit. From [118]. (b) Sketch of the stress response to a shear start-up experiment. The yield stress can be defined as the stress corresponding to the end of the linear regime, as the stress maximum, or as the equilibrium stress. From [16]. (c) Strain response to step stress experiments for various stresses ranging from 0.22 to 220 Pa. From [95]. (d) Oscillatory stress sweep experiment performed on a 6% wt carbon black gel at two different sweep rates: 7 (open) and 34 (filled) $\mathrm{mPa}{\mathrm{s}}^{-1}$. Here the yield stress, defined as the intersection of $G^{\prime }$ (dark gray, blue) and $G^{\prime \prime }$ (light gray, red), depends on the sweep rate. From [330].

Figure 12

(a) Stress relaxation upon flow cessation: experiments with microgels for different preshear stresses (from 60 to 443 Pa). The internal stress ${\sigma }_r$, defined by linear extrapolation of the stress measured over a short time interval ($<50\mathrm{s}$) after flow cessation, is larger for smaller preshear stress and becomes quite significant for preshear stresses approaching the yield stress. From [281]. (b) Evolution of the stress after flow cessation normalized by the stress prior to flow cessation (${\sigma }_{ss}$) as a function of $\dot{\gamma }$ for a hard-sphere colloidal suspension. The curves correspond to various imposed shear rates $\dot{\gamma }$ prior to flow cessation, and different packing fractions. Glass states are shown in light gray (red), liquid states in dark gray (blue). From [8].

Figure 13

(a) Normalized off-diagonal component of the second moment tensor of the dimensionless scattering vector $\widehat{q}$ weighted by the structure factor $⟨\widehat{q}\widehat{q}⟩$, and shear stress ${\sigma }_{xy}$ vs strain $\gamma$ during a shear start-up experiment ($\dot{\gamma }=0.17{\mathrm{s}}^{-1}$) for a polystyrene gel ($\varphi ={10}^{-3}$). (b) Contour plots of a representative cascade of scattering patterns collected during a start-up experiment ($\dot{\gamma }=0.56{\mathrm{s}}^{-1}$) with a $t=0.1$, b 1.1, c 2.2, d 3.5, e 6.3, and f 8.3 s. Maximum anisotropy is observed at $t\simeq 3.5\mathrm{s}$. (a), (b) From [283]. (c) Evolutions of the shear stress and (d) positions of the $T1$ events in a foam sheared in a 2D Couette cell as a function of the applied strain during a shear start-up experiment. Data from numerical simulations. From [207].

Figure 14

Shear rate responses vs time for creep experiments at different imposed shear stresses in various materials: (a) polycrystalline hexagonal columnar phase. From [24]. (b) Carbopol microgel. From [123]. (c) Core-shell PS-p-NIPAM particle glass. From [394]. (d) Carbon black gel at 8% wt. From [402].

Figure 15

(a) Evolution of the shear moduli of a Pickering emulsion stabilized by silica colloids during a LAOS strain amplitude sweep. The volume fraction of the oil is 65%. (b) Confocal images of the emulsion during shear taken 40 mm into the sample to avoid wall effects and obtained at different strains during the strain sweep. Scale bars correspond to $20\mu \mathrm{m}$. For ${\gamma }_0<0.10$, the droplets slide along each other but remain trapped in the cages formed by their neighbors. For ${\gamma }_0\simeq 0.10$, the moduli intersect and the droplets can be seen to move irreversibly, although their displacement over a period is much less than their diameter. For ${\gamma }_0>0.30$, $G^{\prime }$ and $G^{\prime \prime }$ increase due to jamming, which results in apparent shear thickening, and the droplets move rapidly during each period, over distances larger than their own diameter. From [185].

Figure 16

(a) Stress $\sigma$ vs the apparent shear rate ${\dot{\gamma }}_{\text{app}}$ for a microgel paste (open and solid circles) and an emulsion (open and solid triangles) of packing fraction $\varphi \simeq 0.77$ obtained in a cone-and-plate device for smooth (open symbols) and rough (closed symbols) surfaces. Regimes I–III refer to microgel slip behavior discussed in the text. The inset shows the velocity profile measured with rough surfaces for $\sigma /{\sigma }_y=1.05\pm 0.1$. (b), (c) Velocity profiles measured with smooth surfaces for (b) $\sigma /{\sigma }_y=0.9\pm 0.1$ and (c) $1.3\pm 0.1$. Adapted from [270].

Figure 17

(a) Slip velocity vs excess stress in a dense emulsion for stresses below the yield stress in a plate-plate geometry. The top plate is coated with either a weakly adhering polymer surface (open and solid squares) or a nonadhering glass surface (open and solid circles). The wetting properties of the boundary conditions strongly impact the behavior of the slip velocity. From [387]. (b), (c) Slip velocity vs shear stress at the rotor (solid circles) and stator (open circles) for (b) a dense emulsion $\varphi =0.75$ and (c) a dilute emulsion $\varphi =0.2$. The slip velocity is linear in the dilute regime and quadratic for dense packing for stresses above the yield stress. From [371]. (d) Slip velocity vs shear stress in a suspension of ammonium sulfate particles in poly(butadiene acrylonitrile acrylic acid) terpolymer (PBAN). Data obtained in capillary flows with dies (extrusion nozzles) of various aspect ratios (open symbols) and a plate-plate geometry (stars). The solid line corresponds to a linear behavior. From [441].

Figure 18

Shear stress vs shear rate for a dense emulsion ($\varphi =0.923$) measured in a plate-plate geometry with smooth boundary conditions for two different gap sizes (500 and $750\mu \mathrm{m}$, respectively). The stress vs the shear rate computed from the method developed by Mooney and extended by Yoshimura and Prud’homme displays a yield stress, while this is not obvious from the raw measurements. From [442].

Figure 19

Velocity profiles of a dense emulsion flowing in a rectangular microchannel (gap $w=400\mu \mathrm{m}$). Images obtained by confocal microscopy. Velocity profiles in blue (dashed lines) correspond to smooth boundary conditions treated with a piranha solution. The oil droplets experience wall slip. Velocity profiles in red (continuous lines) correspond to smooth, silanized boundary conditions. The oil droplets stick to the surface, creating an effectively rough boundary condition. Flow rates: 0.2, 0.5, and $1.2\times {10}^{-2}\mathrm{mL}/\min$. From [323].

Figure 20

Representative flow curves for (a) a simple yield stress fluid, here microgels of different cross-link densities and concentrations. The solid line is the equation $\sigma /{\sigma }_y=1+(\dot{\gamma }{\tau }_{\beta }/{\gamma }_0{)}^{0.45}$, where ${\tau }_{\beta }$ is the fluid relaxation time. From [82]. (b) Three different thixotropic materials. From top to bottom: a hair gel, a commercial mustard, and a bentonite suspension. Note that each of these materials displays a minimum shear rate ${\dot{\gamma }}_c$ below which no steady flow is possible. The horizontal dotted lines indicate the yield stresses of the different materials. From [95].

Figure 21

Temporal evolution of the apparent viscosity $\eta =\sigma /\dot{\gamma }$ of a drilling mud for various applied shear stresses below and above the yield stress ${\sigma }_y\simeq 2.82$ Pa. The drastic change of behavior within a range of less than 0.1 Pa around the yield stress illustrates the viscosity bifurcation scenario. From [354].

Figure 22

Example of shear-banded flows. (a) Velocity profiles in a 4° cone-and-plate geometry of a colloidal silica suspension (Ludox TM-40) for shear rates ranging from 15 to $105{\mathrm{s}}^{-1}$. (b) Steady-state flow curve determined by two different types of measurements. The branch at larger shear rates is obtained under constant external stress. The branches at lower shear rates are determined by estimating the minimum (respectively, maximum) shear stress with (respectively, without) flow. From [288].

Figure 23

Dimensionless flow curves (stress $T$ vs shear rate $\mathrm{\Gamma }$) for different values of the ratio $D$ of the fluid relaxation time ${\tau }_{\mathrm{rel}}$ over the restructuration time ${\tau }_{\text{age}}$, i.e., the time for a microscopic link to reform after being broken. From [93].

Figure 24

Scenario for shear banding in yield stress materials. A monotonic flow curve with a finite dynamic yield stress ${\sigma }_y$ coexists with a static branch at $\dot{\gamma }=0$ and $\sigma <{\sigma }_y^s$, where ${\sigma }_y^s$ is the static yield stress. The shear rate is bi-valued for a range of shear stresses $\sigma \in [{\sigma }_y,{\sigma }_y^s]$, which may lead to shear bands. Adapted from [31].

Figure 25

Global and local flow curves (black solid line and symbols, respectively) for a dense emulsion ($\varphi =0.75$ and 20% polydispersity). Global data are obtained in a wide-gap Taylor-Couette cell, while local flow curves are deduced from velocity profiles measured in a $w=250\mu \mathrm{m}$ thick microchannel with rough surfaces, for various pressure drops ranging from 300 to 900 mbar (inset). No overlap of the local flow curves is observed. Dashed lines are predictions for the local flow curves at the given pressure drop, as obtained from the nonlocal rheological model [see Eq. (10)] with a flow cooperativity length $\xi =22.3\mu \mathrm{m}$. Inset: solid lines are the velocity profiles predicted by the nonlocal rheological model. The $y$ axis for the main figure is the stress in Pa and the figure thus represents the flow curve; the important observation is that the flow curves for different driving pressures no longer overlap due to collective effects. From [168].

Figure 26

Shear-induced patterns observed in various yield stress fluids under shear in confined geometries. (a) Carbon black gel under simple shear with a gap thickness of $173\mu \mathrm{m}$ as seen with optical microscopy with large and low magnifications (left and right, respectively). From [171]. (b) Suspension of microfibrillated cellulose at 0.1% wt after shearing 10 min at $0.5{\mathrm{s}}^{-1}$ in a Taylor-Couette cell. From [210]. (c) Emulsions under simple shear for a gap thickness of $12\mu \mathrm{m}$. The arrow indicates the direction of shear. From [289]. (d) Optical micrograph of a quiescent semidilute non-Brownian colloidal nanotube suspension at 0.5% wt. The gray (red) scale bar is $10\mu \mathrm{m}$ and the gap thickness is $50\mu \mathrm{m}$. From [244].

Figure 27

Transient shear banding in a Carbopol microgel in the Taylor-Couette geometry. (a)–(e) Velocity profiles $v(r)$, where $r$ is the distance to the rotor, in a rough geometry at different times during the stress relaxation for an applied shear rate of $0.7{\mathrm{s}}^{-1}$. From [128]. (f) Spatiotemporal diagram of the local shear rate $\dot{\gamma }(r,t)$ in a smooth geometry for an applied shear rate of $0.5{\mathrm{s}}^{-1}$. The white curve traces the position $\delta (t)$ of the interface between the fluidized band and the solidlike region. The vertical dashed line indicates the fluidization time ${\tau }_f$, i.e., the time at which the shear rate field becomes homogeneous. From [129].