Length scales and self-organization in dense suspension flows

Dense non-Brownian suspension flows of hard particles display mystifying properties: As the jamming threshold is approached, the viscosity diverges, as well as a length scale that can be identified from velocity correlations. To unravel the microscopic mechanism governing dissipation and its connection to the observed correlation length, we develop an analogy between suspension flows and the rigidity transition occurring when floppy networks are pulled, a transition believed to be associated with the stress stiffening of certain gels. After deriving the critical properties near the rigidity transition, we show numerically that suspension flows lie close to it. We find that this proximity causes a decoupling between viscosity and the correlation length of velocities $\xi$, which scales as the length $l_c$ characterizing the response to a local perturbation, previously predicted to follow $l_c\sim 1/\sqrt{z_c-z}\sim p^{0.18}$, where $p$ is the dimensionless particle pressure, $z$ is the coordination of the contact network made by the particles, and $z_c$ is twice the spatial dimension. We confirm these predictions numerically and predict the existence of a larger length scale $l_r\sim \sqrt{p}$ with mild effects on velocity correlation and of a vanishing strain scale $\delta \gamma \sim 1/p$ that characterizes decorrelation in flow.

Figure 1

(a) Random floppy network under shear deformation approaching the SIJ transition; the thickness of the lines connecting the particles is proportional to the tension (red) or compression (black). Arrows indicate the direction of the shear. (b) Snapshot of a contact force network in the ASM of suspension flow, close to but below the jamming threshold. (c) Jamming phase diagram of networks in the coordination deficit $\delta z\equiv z_c-z$ and shear strain $\gamma$ plane, showing the line of the shear-induced transition (red solid line) ${\gamma }_c\sim \delta z$ [20]. Our contention is that the contact networks in dense suspension flows lie very close to this stiffening line, thus defining a characteristic strain $\delta \gamma$ vanishing at jamming. (d) One-dimensional chain under an extensional flow at four different times. The vertical position of the chain has been shifted for clarity; the upper curve represents the initial configuration. (e) An almost straight chain buckling under compressive flow.

Figure 2

Properties of a one-dimensional chain stretched by an extensional flow with $N=10000$ under periodic boundary conditions (see Appendix pp4 for details on the numerical methods used). The initial distribution of the angles is random (i.e., $\mu =0$). (a) $\langle V^2\rangle =\tau$ versus strain $\gamma$. The blue squares describe the evolution toward the strain-induced jamming point and the red circles describe the buckling transition. The reversibility of the backward evolution breaks at some strain ${\gamma }_{\mathrm{noise}}$. The only source of noise is roundoff error. (b) A log-log plot of $\langle V^2\rangle$ versus the distance to the critical point $\Delta \gamma$. The predicted exponent are 3 and 1, in perfect agreement with the simulations. (c) Average response to a local perturbation $\langle {\tau }_n\rangle$ vs the distance $n$ to the perturbation. (d) Rescaled average response $\langle {\tau }_n\rangle$ vs rescaled distance $n/l_r\sim n\Delta {\gamma }^{3/2}$. (e) Average velocity correlation $C\left(n\right)$ vs the distance $n$. The inset shows the rescaled average velocity correlation $C\left(n\right)$ vs rescaled distance $n/l_{\mathrm{corr}}\sim n\Delta \gamma$.

Figure 3

Two-dimensional network, built as in [20], under shear. (a) Stress $\sigma$ vs strain $\gamma$. The blue circles represent the dynamics towards the SIJ transition, whereas the red squares represent the buckling transition. (b) Scaling of $\frac{d\sigma }{d\gamma }/\delta z^{0.5}$ vs $\sigma$ in the vicinity of the buckling transition (see Appendix pp4 for details about numerical methods). The slope of the line is the predicted exponent 2.

Figure 4

Spectral density $D\left(\omega \right)$ of the $\mathcal{N}$ operator for two-dimensional networks of size $N=4096$, measured numerically for $\delta z=0.4$ at various stresses, as indicated in the legend. In undeformed networks a gap is observed in $D\left(\omega \right)$ up to a frequency ${\omega }^{*}\sim \delta z$ [15]. Above ${\omega }^{*}$, $D\left(\omega \right)$ remains effectively unchanged under the deformation, whereas below ${\omega }^{*}$ a single mode of frequency ${\omega }_{\min }$ rises from zero as $1/\sqrt{\sigma }$ (at constant $z$) upon buckling, for which $\sigma \to \infty$. Notice that $D\left(\omega \right)$ displays modes between ${\omega }_{\min }$ and ${\omega }^{*}$ indicated by the arrow; these are plane-wave modulations of the minimal mode and thus scale like $1/L$.

Figure 5

(a) Velocity field amplitude $V\left(r\right)$ as a response to a local perturbation vs the distance to the perturbation $r$ for different shear stress in two dimensions. (b) Considering the predicted length scale $l_r\sim \sqrt{\sigma }$, the rescaled velocity $V(r/l_r)l_r$ vs rescaled distance $r/l_r$ collapse on a single curve for a fixed coordination in agreement with the theory. The inset shows the total velocity field displacement $\langle V|V\rangle$ vs the stress $\sigma$, showing the predicted logarithmic increase.

Figure 6

(a) Typical stress vs strain signal from the ASM simulations [12, 26]. The stress relaxes between the abrupt increases that signal collisions of the sheared hard particles. The slope of the stress during the relaxation segments depends on the stress itself as $d\sigma /d\gamma \sim -{\sigma }^2$, as derived in Appendix pp3 and validated numerically in Fig. 7. (b) Illustration of the relation between the stress relaxation in between collisions in the ASM and the buckling transition: The stress follows $\sigma \sim {(\gamma -{\gamma }_c)}^{-1}$ (red solid curves) between collisions, which is analogous to the stress relaxation of floppy networks evolving away from a buckling transition. The dashed vertical lines indicate the strain ${\gamma }_c$ at which a floppy network, constructed by connecting the centers of the hard particles in contact by rods, would undergo a SIJ transition if it were sheared backward. We note that while the strain intervals between collisions (the length of the dotted curves) vanish in the thermodynamic limit, the strain scale $\delta \gamma \equiv \gamma -{\gamma }_c\sim 1/\sigma$ is independent of system size.

Figure 7

Hard-particle suspensions modeled by the ASM simulated as in [12]. (a) Distance in strain $\delta \gamma$ to the SIJ point (as explained in the text) vs the stress $\sigma$. (b) Scaling of $-\frac{d\sigma }{d\gamma }$ vs $\sigma$ for hard-particle systems under shear flow. (c) Average velocity response to a local perturbation $V\left(r\right)$ vs the distance $r$ to the perturbation. (d) Average response $V\left(r\right)$ vs the rescaled distance $r/l_c$. (e) Velocity correlation $C\left(r\right)\equiv \langle {\vec{V}}_i·{\vec{V}}_j\rangle /\langle V^2\rangle$ vs the distance $r$ between particles $i$ and $j$. (f) Velocity correlations $C\left(r\right)$ vs the rescaled distance $r/l_c$.

Figure 8

Velocity correlations $C\left(r\right)$ for networks under shear. (a) The raw correlation functions have an initial exponential decay on the length scale $l_c\sim 1/\sqrt{\delta z}$. We also show the velocity correlations measured in an undeformed isotropic network of the same coordination (black dots), demonstrating that the initial decay occurs on the same length scale, independent of the stress. (b) The correlation functions rescaled by constants $b_{\sigma }$, which are chosen such that the functions collapse with the second length scale $l_r$, scaling as $l_r\sim \sqrt{\sigma }$ at fixed $z$. We find numerically $b_{\sigma }\sim ln\left(\sigma \right)$. In the inset of (b) we plot the velocity correlations measured in undeformed isotropic networks with various coordinations.