Dense suspensions as trainable rheological metafluids

Abstract

In materials, the ability to retain the memory of applied stresses or strains opens up new opportunities for enhancing their performance adaptively via training. In dense suspensions, a stress-adaptive response is enabled by non-Newtonian rheology; however, typical suspensions have little memory, which implies rapid cessation of any adapted behavior. Here, we show how multiple adaptive responses can be achieved by designing suspensions where different stress levels trigger different memories. This is achieved through the interplay of particle interactions based on frictional contact and dynamic chemical bridging. These two interactions give rise to stress-activated memories associated with opposite time-dependent trends. As a result, a suspension can be trained to adapt to applied stress either by softening or stiffening, exhibiting targeted viscosity and energy dissipation in response to low-velocity impact. Such behavior, usually associated with mechanical metamaterials, suggests that dense suspensions with multiple memories can be viewed as trainable rheological metafluids.

Keywords: dense suspensions; metafluids; non-Newtonian rheology.

Fig. 1.

Design and rheology of trainable suspensions with multiple memories. (A) Trainable suspensions can achieve varying viscosity states through training protocols. These protocols enable memory formation (M), allowing for an extended lifetime of shear-induced microstructural states. Introducing multiple memory-forming units (M₁, M₂, …, and M_n) enables trainable material properties, thereby tailoring viscosity states. The trained states are persistent until they relax back. (B) Schematic illustration of the metafluid design. Fumed silica particles are surface-functionalized with thiol groups (OX50-SH, red triangles) and suspended in a NO₂-BCAm-endcapped poly(propylene glycol) Michael-acceptor (concave pentagons). Two distinct interparticle interactions are induced under shear: 1) chemical friction via the thia-Michael reaction between surface thiols and NO₂-BCAm motifs and 2) particle–particle frictional force. The Inset shows a transmission electron microscopy image of thiol-functionalized fumed silica particles. (C) Rheology of the OX50-SH in NO₂-BCAm-endcapped poly(propylene glycol) Michael-acceptor suspensions at weight fractions ${\varphi }_{\mathrm{w}}=35$, 38, and 40% (Bottom to Top). Viscosity η is measured by the forward (closed symbol) and backward (open symbol) ramp of applied shear stress τ. The mean and SE shown in (C) are based on three replicates. The shaded area is the inaccessible window that reaches minimum shear rates measurable by the rheometer.

Fig. 2.

Hysteresis and time dependence of shear rheology. (A) Viscosity of the ${\varphi }_{\mathrm{w}}=40$% suspension in Fig. 1C for different stress-controlled ramp rates, where 5 s (circles), 30 s (squares), or 90 s (triangles) were allotted to measuring each data point. Only forward ramps are shown. (B) Hysteresis loops for different stress ranges. The colored area is the range of τ measured. Arrows indicate the direction of the stress sweep.

Fig. 3.

Training via repeated vertical impact. (A) Setup for the impact test. For the training process, the impacting rod travels $L=5$ mm up and down at a fixed velocity $v_{\mathrm{train}}$ and the normal force $F_{\mathrm{N}}$ on the impactor is tracked as a function of time. (B) For low $v_{\mathrm{train}}$, stiffening of the suspension is observed. (C) The suspension softens when repeated impact occurs with larger $v_{\mathrm{train}}$. (D) Effective impact resistance $R_{\mathrm{eff},\mathrm{n}}$ during training as a function of training cycle n, normalized by $R_{\mathrm{eff},1}$ for the first cycle. (E) $R_{\mathrm{eff},\mathrm{n}}$ for the first cycle ($n=1$, closed symbols) and the final cycle ($n=10$, open symbols) as a function of $v_{\mathrm{train}}$. Red horizontal lines in (B and C) refer to the maximum $F_{\mathrm{N}}$ during the 10 training cycles.

Fig. 4.

Read-out of memories in the trained metafluid. (A) Illustration of the read-out protocol (blue background) after 10 cycles of training. After training at given $v_{\mathrm{train}}$ (yellow background), the impactor rod compresses the trained fluid at fixed $v_{\mathrm{read}}=0.1$ mm/s while the normal force $F_{\mathrm{N}}$ is measured. (B) Read-out force for low impact speeds $v_{\mathrm{train}}=0.03$, 0.04, 0.1, and 0.2 mm/s (Top to Bottom). (C) Read-out force for larger impact speed $v_{\mathrm{train}}$= 0.5, 1, 2, and 30 mm/s (Top to Bottom). (D) Effective impact resistance during read-out $R_{\mathrm{eff},\mathrm{read}}$ at the impact depth $L=4$ mm and (E) energy absorbed $E_{\mathrm{abs}}$ during read-out with constant read-out velocity $v_{\mathrm{read}}=0.1$ mm/s after being trained at varying training velocities $v_{\mathrm{train}}$. (F) Lifetime of memories trained at $v_{\mathrm{train}}=1$ mm/s. The mechanical response of the trained suspension is measured at the same compression speed of 1 mm/s after a relaxation time of 0, 5, 10, 30, 100, and 300 s. (G) Peak normal force during read-out $F_{\mathrm{N},\mathrm{read}}^{\max }/F_{\mathrm{N},1}^{\max }$ normalized by the peak force of the untrained suspension $F_{\mathrm{N},1}^{\max }$ as a function of relaxation time.