Soft materials exhibit various morphological features by virtue of their ready deformability. Often, mechanical interactions in soft materials cause surface buckling instabilities, which manifest periodic wrinkles, localized creases, or folds [1,2]. These instabilities can be of significant interest for applications in engineering, such as controlled optical properties [3,4] or predicting buckling modes of thin films [5], and also pose challenging theoretical questions.

Surface instabilities have been studied primarily in the context of compression-driven instabilities, which arise from processes such as biological growth [6,7], swelling [8,9], or friction [10,11]. Studies on elastic surface instabilities typically have involved soft solids whose shear moduli were Oð10 3 Þ Pa or larger.

Recently, the mechanics of solids in an extremely soft limit, whose equilibrium storage moduli G 0 ¼Oð0.1-10ÞPa, have gained attention, owing to their natural connection to soft, solidlike systems found in biology [12][13][14]. Because of the loosely connected network structure, these solids exhibit rate-, time-, and strain-dependent elasticity; similar rheological characteristics are found in imperfectly networked elastomers [15,16]. Consequently, understanding the mechanical traits of these synthetic and biological ultrasoft solids is essential.

This Letter presents a surface instability observed during the movement of ultrasoft solids within confinement; we are not aware of previous reports of the phenomenon documented here. We demonstrate that these solids can translocate through narrow gaps by continuously inverting their own body inside out. This deformation process leads to the emergence of furrowlike surface grooves, forming primarily due to the solid gel's "inside out" deformation, and distinct from previously known types of surface instabilities. We confirm this distinction theoretically and experimentally and offer a scaling analysis to explain the cause of the "furrowing" instability.

In our experiments, two acrylic cylinders, one hollow and the other solid, machined and surface polished, were brought together to create a hollow annular channel of different widths d ¼ 2w; the centerline radius of the hollow annular channel is denoted R c [Fig. 1(a)]. The outer radius of the annulus was R o ¼ R c þ w ¼ ½7.9; 12.6; 15.5; 21.9; 48 AE 0.3 mm, and we adjusted the radius of the inner cylinder to give eleven different values of w ¼ ½0.40;0.65; 0.90; 1.15; 1.40; 1.65; 1.90; 3.10;3.60;4.55; 14.5 AE 0.08 mm. Axisymmetric geometries were explored primarily due to having continuous boundaries to consider, whereas channels with rectangular cross section introduce additional boundaries and corner effects; future experiments with long and narrow rectangular geometries may provide new insights. The viscoelastic gel material was cast and left to cross-link in the "reservoir" below the annulus [Fig. 1(a)].

The viscoelastic gel was composed of vinyl-terminated polydimethylsiloxane (PDMS) (viscosity μ ¼ 4850 mPa s; Gelest) and silicone oil (μ ¼ 20.8 mPa s; Sigma Aldrich). The vinyl group on PDMS reacts with a trimethylsiloxane terminated (15%-18% methylhydrosiloxane) dimethylsiloxane copolymer (Gelest) to form a cross-linked gel, and platinum-divinyl tetramethyldisiloxane complex 2% Pt in xylene (Gelest) was used as the catalyst [17]. All materials were used as purchased. We explored three different crosslinking concentrations, c ¼ ½0.425; 0.45; 0.5 wt% of the total weight of the PDMS and silicone oil mixture.

The network components were mixed at nonstoichiometric ratios, thus creating dangling chains [18]. Dangling chains produce time-dependent (viscoelastic) material response in cross-linked gels. On short timescales, dangling chains entangle with the existing network, providing higher elastic resistance. On longer timescales, entangled dangling chains relax while the chemically cross-linked network provides the baseline elastic resistance [15,19] (see Supplemental Material [20], Sec. I). The storage modulus G 0 in the long-time limit, which we denote G 0 , was Oð0.1-10Þ Pa.

Once the gel was fully cured, the gel was pushed vertically upward at a constant speed to flow (deform) into the hollow annulus (see Supplemental Material [20], Sec. VIII). The position of the gel front l was measured relative to the beginning of the annulus. The ratio l=w provides a measure of axial strain and the average strain rate can be defined as ε ¼ l=ðwtÞ, where t is time, with t ¼ 0 when the gel first enters the annulus. ε was set by controlling the pump (PHD-Ultra, Harvard Apparatus), which was used as a linear translation stage. Once the gel front reached a certain length from the beginning of the annular section, the free surface of the gel spontaneously deformed, showing a periodic morphology [Fig. 1(b)] (see Ref. [20], Sec. II). We denote this critical length as l c for the first appearance of this instability. The instability displayed larger morphological features with increasing w=R c , as observed with top views of the gel [Fig. 1(c)]. During the entire length of extrusion, no sign of gel fracture was observed.

We first discuss the onset characteristics of the instability. The critical length l c was measured at varying cross-linker concentration c, strain rate ε, the half-gap w, and the outer radius R o [Fig. 2]. l c did not vary between different combinations of c and ε, however, it was most strongly affected by the geometrical length scale w. Within the range of w studied, l c ∝ w 4=3 . We explored the role of the confinement geometry more extensively and identified that l c ∝ R -1=3 c ; thus we could organize all the data as l c ∼ w 4=3 R -1=3 c [Fig. 2(a)] (see Ref. [20], Sec. III). While the surface morphology of the instability shown in Fig. 1 resembles those observed in the compression-driven crease instability, we elucidate several key differences that set the instability documented here apart from the crease instability. The surface crease instability appears when a soft solid is compressed laterally greater than ε crease ¼ 35.4% of its initial length [25], or more if the effect of surface tension γ is significant. Localized creases are naturally aperiodic, but when the two length scales, the gel thickness w, and the elastocapillary length γ=G 0 are comparable, periodically buckled surface pattern can form [26]. In such cases, the wavelength and the critical compressive strain depends on both w and γ=G 0 [27]. Considering that the scale of the elastocapillary number that we investigated ranged γ=wG 0 ¼ Oð10 -2 -10 2 Þ, critical compressive strains, if we are observing a creaselike instability, would need to be at least 35% or be larger than 75% [26]; however, compressive strains on the gel free surface in our annular geometry are less than 30% and typically smaller, as obtained by comparing the ratio of circumferences along R c and R i ¼ R cw. Our experimental observations of the instability indicate distinct features from the crease instability. First, we showed that the instability still appeared without the gel reservoir, or even without the annular confinement, such that there is no overall compression of the elastic gel in the radial and circumferential directions (see Ref. [20], Sec. III). To show that, we experimented with hollow cylinders, not annulus, of varying outer radii R o ¼ R c þ w ¼ ½3.8; 5.2; 8.0; 10.8; 15.5 AE 0.3 mm. For hollow cylinders, we treated R i ¼ R cw → 0, and therefore, we examined if l c ∼ w 4=3 R -1=3 c ∼ w. As expected, measurements showed l c ∼ w, confirming that compression is not the driving mechanism for this instability. Furthermore, our experiments show that both the critical transverse strain l c =w ∼ ðw=R c Þ 1=3 and the instability wavelength λ ¼ 2πR c =N ∼ w depend only on the geometrical length scales w and R c , rather than being affected by γ=G 0 [Fig. 2].

We suspected that material confinement, i.e., interactions with the sidewalls, was a potential cause of the instability. Hence, we performed experiments after lubricating the sidewalls with a thin layer of silicone oil (μ ¼ 5 mPa s). Such manipulation allowed the gel entering from the reservoir to slip on the wall of the annulus. Nevertheless, the gel experiences radial and circumferential compression as it deforms into a smaller area. The critical length for the instability increased to the prediction of the crease instability, which we confirm by performing a linear perturbation analysis (see Ref. [20], Sec. IV).

To explore the deformation of the solid gel within confinement, we mixed density-matched polystyrene beads (ρ ¼ 0.96 g=cm 3 , 106-125 μm diameter; Cospheric), and then tracked the position of the particles [Fig. 3(a)]. Once the surface was deformed, beads positioned underneath the furrowing folds stalled. As the gel progressed further into the annulus, some furrows grew, increasing λ. During this process, beads approaching the dead-end turned away and flowed into surviving, larger protuberances.

The material trajectory of the elastic elements inside the curved tip region is similar to that of the fountain flow in viscous liquids, characterized by rotating toroidal vortices [Fig. 3(b)]. We call this type of solid deformation "eversion." This observation evinces that soft solids can move in confinement by constantly turning their body inside out. We note that the instability begins and propagates from the inner boundary, not at the apex of the extrusion front [Fig. 3(c)]. Also, the propagation of the instability happens much faster than the timescale of the extrusion (ε -1 ). This finite-time propagation is attributable to the gel viscoelasticity [25].

The everting path lines of the elastic gel element suggest that the instability observed here is induced by shear driven bending, rather than compression driven. This leads to the conclusion that the instability differs from the crease instability and represents a phenomenon that has not been explored before, at least to our knowledge. Henceforth, we refer to the instability studied in this Letter as the "furrowing" instability, drawing inspiration from its morphological resemblance to surface undulations that progressively deepen over time. Having this knowledge, we relate the relevant variables that characterize the eversion of soft solids to explain the observed power-law relationships; l c ∼ w 4=3 R -1=3 c and λ ∼ w. First, we experimentally observed that the height of the curved front h linearly scales with w; h ≈ 0.8w, independent of ε and c [see Fig. 1(b) to find h]. This relationship is rationalized by modeling the progressing gel "column" as a one-dimensional structure where the addition of the new material is localized in a top region with a constant height h during extrusion [28] (see Ref. [20], Sec. V). Therefore, we consider the curvature of the gel front to scale with w -1 . The advancing gel front features two rotating (elastic) vortices within the curved tip, with path lines following poloidal trajectories around the meridian. Because of the elastic nature of the material, we conceptualize the rotating vortices as two bending thin (∼w) and wide (∼R c ) sheets. To arrive at the tip, elastic materials experience shear interaction between the upward-flowing (away from the wall) and downwardflowing (near the wall) regions, accumulating the shearinduced strain energy U s . Estimating the scale of strain as ε ∼ l=w and the sheared volume as V ∼ lwR c , we obtain

The confinement length scale governs the scale of bending energy. Indicating the length of the meridian s ∼ wθ, and the bending angle θ ¼ Oð1Þ, we write the bending energy as U b ∼ BAκ 2 , where the bending stiffness B ∼ G 0 w 3 , area A ∼ sR c , and curvature κ ¼ ∂θ=∂s ∼ θ=s, from which we obtain U b ∼ G 0 w 2 R c . The scale of the bending energy does not increase with l as s is constant during the progression of the tip.

In the case of eversion inside an annulus, we consider the interplay of two principal curvatures (κ 1 ∼ w -1 and κ 2 ∼ R -1 c ) introduced by the poloidal-toroidal geometry of the front. The material element moving along the poloidal trajectory experiences a sign change in the Gaussian curvature, which scales as κ G ¼ κ 1 κ 2 ∼ ðwR c Þ -1 . Therefore, we write a Gaussian bending energy U b;G ∼ BAκ G ∼ G 0 w 3 [29], and thus the total energy is U b þ U b;G . In contrast, the scale of the bending energy in the fully developed instability state, as sketched in Fig. 3(d), with the furrow depth a ∼ λ, can be written accounting for both Gaussian and mean curvatures [κ 02 ∼ w -2 þ λ -2 þ ðwλÞ -1 ] as NBA 0 κ 02 ∼ G 0 w 2 R c ∼ U b , where A 0 ∼ sλ, N ∼ R c =λ, and λ ∼ w (experimentally observed). Hence, we conclude that the furrowing instability lowers the total energy by ∼U b;G by breaking up a long (∼R c ), rotating curved surface into smaller (∼λ) protuberances.

We then identify the critical extrusion length l c from where U s and U b;G become comparable; in other words,

. For l < l c , the energetic cost associated with azimuthal bending dominates shear energy. Hence, surface elements are stable against perturbations, as plane shear extrusion is energetically favorable against bending in the azimuthal direction. If instead l > l c , continued extrusion costs more energy as U s ∝ l 3 , and, therefore, azimuthal bending due to perturbations becomes likely as it constitutes a reduced total energy.

As the gel progresses by rotating the material element along the free surface, the position of the gel element in the comoving frame of reference [Fig. 3(b)] can be characterized by angular position θ. Therefore, emergence of the instability involves a perturbation to this degree of rotation, δθ [Fig. 3(e)]. This angular perturbation δθ produces torsional energy δU t within the gel front, δU t ∼ G 0 Jδθ 2 =L ∼ G 0 w 2 δa 2 =λ, where the torsional moment of inertia scales J ∼ w 4 , the length scale of torsion scales L ∼ λ, emerging furrow depth at the gel-wall contact line scales as δa ∼ wδθ. Likewise, δa causes azimuthal bending δU b ∼ G 0 w 5 δa 2 =λ 4 . As the two energy scales must balance in order to minimize the perturbed state of the energy, we balance

and arrive at λ ∼ w.

As the extrusion is continued, we measured the wavelength λ and the furrow depth a for different values of c and ε, as a function of the excess strain defined as ðl -l c Þ=w [Fig. 4; Ref. [20], Sec. VI]. Regardless of c and ε, the furrows evolve longer wavelengths once ðl -l c Þ > w. The growth of furrow height a½lðtÞ depended on the elasticity of gels [Fig. 4(a); see Fig. 1(b) to find a]. We observed no strong dependence of a on ε, which supports our conclusion that the instability is elasticity driven.

The observed phenomenon has significance for both industrial and biological contexts. For example, coalescence of the furrows engulfed trails of air pockets following behind the advancing gel front [Fig. 4(b)]. Such air pockets can manifest as defects in objects manufactured through the extrusion of viscoelastic melts [30]. Additionally, the rheological characteristics of the soft solids investigated this Letter bear qualitative resemblance to of crosslinked networks found within cellular bodies. Consequently, our Letter may offer qualitative comparison between biological soft solids and synthetic soft solids moving inside confinement.

In conclusion, our Letter demonstrates the remarkable deformability of soft solids, showcasing their ability to flow into narrow gaps by continuously everting their bodies through shear interaction with the confining walls. Prolonged eversion of the thin annular soft body resulted in a novel form of surface instability, we termed the "furrowing" instability. Our experiments showed that the critical strain and the instability wavelength depend only on the geometric length scales. These findings shed light on fundamental aspects of material behavior and also carry implications for both industrial processes and biological systems, offering insights into the motion and stability of soft bodies within confinement.