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The chain fountain phenomenon, also known as the Mould effect, self-siphoning beads, or Newton's beads, is a physical phenomenon observed with a chain placed inside a jar. One end of the chain is pulled from the jar and is allowed to fall under the influence of gravity. This process establishes a self-sustaining flow of the chain which rises over the edge and goes down to the floor or ground beneath it, as if being sucked out of the jar by an invisible siphon. For chains with small adjacent beads, the arc can ascend into the air over and above the edge of the jar with a noticeable gap; this gap is greater when the chain falls farther.[1]

The self-siphoning effect is also observed in non-Newtonian fluids.

History

[edit]

The self-siphoning phenomenon has been known for some time, and had become a topic of public discussion many times in the past. Science entertainer Steve Spangler presented this phenomenon on TV in 2009, both with beads and viscoelastic liquids.[2][3] This phenomenon is classically known as Newton's beads.[4]

The effect is most pronounced when using a long ball chain. The higher the jar containing the chain is placed above the ground, the higher the chain will rise above the jar during the "siphoning" phase. As demonstrated in an experiment, when the jar is placed 30 metres (98 ft) above the ground and the chain is sufficiently long, the arc of the chain fountain can reach a height of about 2.1 m (6 ft 11 in) above the jar.[non-primary source needed][5]

In 2011, the phenomenon with the rising chain was described as an open problem[a] for the 2012 International Young Physicists' Tournament (IYPT).[6][4]

In 2013, science presenter Steve Mould (namesake of the effect) brought the chain problem to widespread public attention.[7] In a YouTube video,[8] he demonstrated the phenomenon of self-siphoning rising beads, and he then presented his proposed explanation on a BBC show.[9] This publicity caught the attention of academics John Biggins and Mark Warner of Cambridge University.[10][11]

In 2014, Biggins and Warner published their findings on the problem in Proceedings of the Royal Society, regarding what they called the "chain fountain" or the "Mould effect".[12][13][14][7]

Explanation

[edit]
Slow-motion video showing chain fountain dynamics.

A variety of explanations have been proposed as to how the phenomenon can best be explained in terms of kinematic physics concepts such as energy and momentum.[15][16][17][18]

Biggins and Warner suggest that the origin of the upward force is related to the stiffness of the chain links, and the bending restrictions of each chain joint.[7][12]

Furthermore, because the beads of the chain can drag laterally within the jar across other stationary links, the moving beads of the chain can bounce or jump vertically when they strike the immobile links. This effect contributes to the chain's movement, but is not the primary cause.[12][17]

In non-Newtonian fluids

[edit]

The self-siphoning phenomena can be also observed in viscoelastic fluids[19] that are mainly composed of long polymers, like polyethylene glycol.[2][20]

See also

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Notes

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References

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Bibliography

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Chain fountain

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from Grokipedia
The chain fountain is a counterintuitive physical demonstration in which a or of beads, initially piled loosely in a raised such as a beaker, spontaneously forms an arched "" rising above the container's rim when one end is pulled over the edge and released to fall under . This effect, observed with chains of sufficient length and flexibility, occurs as the falling portion pulls the chain from the pile, but instead of simply draping over the rim, the emerging segments are propelled upward to a height proportional to the container's elevation above the ground. The phenomenon, sometimes called the Mould effect after science communicator who popularized it through a 2013 YouTube video viewed millions of times, challenges intuitive expectations of gravitational flow, appearing to defy as the chain leaps skyward before cascading down. It requires the chain to be at rest in a confined pile initially, with the setup typically involving a beaker at height h1h_1 above the floor, leading to a maximum fountain height h2h_2 where h2∝h1h_2 \propto h_1. While earlier demonstrations predate Mould's video, the modern understanding stems from analyses showing that the motion arises not just from tension pulling the chain upward, but from an additional upward reaction exerted by the stationary pile on the moving beads. This reaction force, proportional to the square of the chain's velocity, originates from the pile's inertia resisting the sudden acceleration of beads as they join the flowing segment, effectively pushing them into motion and sustaining the arch. Theoretical models, developed by researchers including J. S. Biggins and M. Warner, treat the chain as a series of point masses connected by inextensible links, revealing that the fountain persists until the chain's speed reaches a balance where the upward push equals the downward pull. Experimental validations confirm the effect's dependence on chain properties like bead size and pile configuration, with no fountain forming if the chain is laid flat or the pile is too loose. Recent microgravity tests, including a 2025 NASA experiment, have further explored the phenomenon in zero gravity. The chain fountain illustrates principles of classical mechanics, including momentum conservation and inelastic collisions within the pile, and has inspired studies in non-equilibrium dynamics and self-sustained flows.

History and Popularization

Early References

The chain fountain effect traces its origins to 19th-century investigations into the dynamics of falling chains, treated as intriguing puzzles in due to their counterintuitive behavior as variable-mass systems. Early explorations began with around 1850, who discussed such problems in private lectures at as part of preparation, highlighting the challenges posed by chains in motion under gravity where mass changes affect acceleration in unexpected ways. These setups were seen as anomalies akin to siphons, where the falling portion exerts a pulling force on the stationary part, defying simple expectations of . A more formal analysis appeared in 1856 with Peter Guthrie Tait and William John Steele's Treatise on Dynamics, which examined a uniform heavy with one end hanging over a smooth and the other coiled on a table. They derived an acceleration of g/3g/3 for the system, where gg is , attributing the reduced rate to the interaction between the moving and stationary segments—a result that puzzled contemporaries because it implied energy dissipation not immediately apparent in the idealized model. This configuration illustrated the siphon-like anomaly, with the descending "pulling" the coiled portion into motion, much like fluid in a , though without quantitative resolution of the visual arching now associated with the fountain. Arthur Cayley built on these ideas in 1857, publishing in the Proceedings of the Royal Society of London a general for a class of dynamical problems, including falling link by link from a heap at the edge of a table through a hole. Cayley confirmed the acceleration of g/3g/3, emphasizing the selective application of Lagrangian methods to account for the impulsive forces at each link's release, which created the counterintuitive slowdown. These 19th-century treatments, confined to theoretical mechanics texts without widespread experimental illustrations, underscored the effect's puzzling nature but remained obscure curiosities. The phenomenon's dramatic visual appeal—chains seemingly defying gravity by rising before falling—went largely unappreciated until viral video demonstrations in the digital age revived interest.

Modern Demonstrations

In the early , the chain fountain phenomenon gained renewed attention through public demonstrations by educators. In , entertainer showcased the effect in a video demonstration using a string of beads that appeared to rise and loop upward as it was pulled from a container, captivating audiences and highlighting its counterintuitive nature. This presentation, part of Spangler's broader efforts to popularize physics experiments, played a key role in sparking widespread interest among teachers, students, and online viewers by framing it as a mesmerizing display of and motion. The phenomenon received further prominence in 2013 when science communicator featured it in a video titled "The Chain Fountain," which amassed millions of views and introduced the term "Mould effect" to describe the self-siphoning behavior. Mould explicitly linked the demonstration to the classic "Newton's beads" analogy, where a falling of beads resists collapsing fully due to , emphasizing the shared principles of dynamics and drawing parallels to historical puzzles in . This video not only popularized the effect in but also prompted initial scientific curiosity by challenging viewers to explain the upward arch without external forces. Concurrently, the chain fountain was formalized as a research problem in academic circles through its inclusion in the International Young Physicists' Tournament (IYPT), listed as Problem 3: "String of beads." The problem described releasing a long string of beads from a beaker, observing the chain's tendency to form an upward loop due to , and tasked participants with investigating the underlying mechanisms. As an unsolved challenge, it stimulated early discussions and experiments among young physicists worldwide, bridging popular demonstrations with structured scientific inquiry.

Recent Research Milestones

The chain fountain phenomenon gained renewed scientific attention following its popularization through demonstrations by science communicator , which spurred theoretical and experimental investigations. A pivotal milestone came in 2014 with the publication of the first rigorous theoretical model by John S. Biggins and in Proceedings of the Royal Society A, explaining the fountain's formation through the dynamics of motion and momentum transfer from the stationary pile. Building on this foundation, a 2018 study in Frontiers in Physics by E. G. Flekkøy, Marcel Moura, and K. J. Måløy utilized numerical simulations and controlled experiments to elucidate the momentum transfer mechanisms driving the arch-like rise, identifying subtle interactions at the chain's pickup point as key to the effect's persistence. In 2021, Dragos-Victor Anghel revisited the phenomenon in Journal of Physics: Conference Series, analyzing the siphoning dynamics and arguing that chain , rather than purely gravitational influences, primarily sustains the fountain, thereby refining earlier hypotheses on force balances. A significant experimental advancement occurred in 2025 when astronaut Don Pettit conducted a microgravity test of the chain fountain aboard the , in collaboration with ; the results showed the chain maintaining its arched shape and flow without significant gravitational pull, refuting models overly reliant on Earth-bound and confirming the role of intrinsic chain dynamics in the effect.

Phenomenon Description

Experimental Setup

The chain fountain effect is typically observed using a simple apparatus consisting of a , such as a 1-liter plastic beaker or tall pot, elevated above a lower surface like a smooth floor or table. A long , often a nickel-plated with 4.5 mm diameter beads connected by short 2 mm rods, is piled loosely inside the container in a random or patterned configuration to minimize tangling. One end of the chain is draped over the rim of the container and extended downward toward the lower surface, with the setup designed to allow the chain to flow freely upon release without external interference. Key parameters include the total chain length, which ranges from 1 to 5 meters for classroom demonstrations to ensure visibility of the arch formation, though longer chains up to 50 meters are used in detailed studies to achieve pronounced effects. The height of the container above the receiving surface, typically 1 to 4 meters, is crucial to provide sufficient drop for the chain to accelerate and form the characteristic structure, while the initial pile must occupy the container without overflowing the rim. The receiving surface should be flat and unobstructed to allow the chain to accumulate without rebounding or altering the flow dynamics. Variations in the setup often involve the type of , such as rigid-linked chains versus more flexible strings, where the maximum (e.g., 63° for typical chains) influences the stability of the pile and flow. Initial pile configurations can be adjusted, for instance, by distributing the chain at multiple points within a bumpy-bottomed spaced at bead diameters to prevent clumping, ensuring repeatable and tangle-free initiation of the effect. These modifications stem from early demonstrations adapted for controlled observations.

Observed Behavior

In the chain fountain experiment, a long beaded is initially piled loosely in a tall pot or beaker elevated above a lower surface, with one end draped over the rim and allowed to fall freely under to the lower surface. The starts stationary in the , but as the hanging portion accelerates downward, the pile inside begins to lift and flow toward the rim, spontaneously forming a smooth, self-supporting arch that rises above the 's edge before arcing over and connecting to the descending segment. This arch typically reaches heights of 10 to 20 cm above the rim in standard tabletop demonstrations using beaded chains of moderate length, creating a persistent "fountain" shape that maintains its form without any external structural support as the chain continues to uncoil and exit the pot. The visual effect is marked by the chain's segments moving upward in a coordinated, looping , contrasting sharply with the expected of material simply spilling over the edge. The counterintuitive nature of the phenomenon lies in the chain's apparent resistance to gravity, as if "climbing" out of the container against the pull of free fall, rather than collapsing or flowing directly downward. The fountain persists steadily until the chain is depleted from the pot, at which point the motion transitions to ordinary free fall of the remaining length, with no further arch formation.

Underlying Physics

Kinematic Aspects

The chain in a fountain configuration follows a distinct divided into three primary segments. It begins in a stationary pile within the , where the chain links are at rest. From the pile, the chain transitions into a rising segment that lifts upward, undergoing a curved deflection at the rim of the due to the geometric constraint of . This curved portion smoothly connects to a straight falling segment that descends vertically toward the ground, maintaining a consistent linear path thereafter. Velocity profiles along the chain exhibit characteristic variations across these segments. In the falling straight segment, the chain propagates at an approximately uniform speed, reflecting steady downward motion without significant or deceleration. Conversely, in the rising segment, the chain experiences deceleration attributable to the of the links as they are set into motion from rest, resulting in a progressive slowdown during ascent. These velocity changes contribute to the observed arch formation as a kinematic outcome of the . Geometric factors play a crucial role in shaping the overall motion. The radius of curvature at the rim directly influences the height of the rising arch, with a smaller typically leading to a more pronounced lift due to the tighter bend constraining the path. Basic kinematic descriptions model positions as functions of time; for instance, in the rising or lowering phases of similar chain motions, the position vector can be expressed as r=∓(Rcos⁡vt2R,Rsin⁡vt2R,v2t2)\mathbf{r} = \mp \left( R \cos \frac{v t}{2R} , R \sin \frac{v t}{2R} , \frac{v^2 t}{2} \right), where RR is the , vv is the characteristic speed, and tt is time, illustrating the parametric evolution of the .

Dynamic Forces

In the chain fountain, the primary dynamic forces governing the motion include , internal tension within the chain, and contact forces at the pot's rim. Gravity provides the driving by pulling the chain downward, while tension transmits along the chain links. Contact forces at the rim, arising from interactions between the chain and the container edge, redirect the chain's path and contribute to the upward deflection observed in the fountain. These forces interact to produce the characteristic arching behavior without violating conservation laws. Tension propagates along the chain with distinct variations between segments. In the falling segment, tension is relatively high, particularly near the top where it must support the weight of the descending chain and provide the to accelerate the system, decreasing toward the bottom due to the cumulative effect of on the length below. In contrast, tension in the stationary pile is low or negligible, as the chain there experiences minimal motion and primarily rests under its own weight until beads are successively engaged. This in tension ensures smooth of motion from the pile to the falling part. At the rim, the chain undergoes redirection as it transitions from the horizontal pile to the vertical fall, involving normal and frictional contact forces from the pot edge. The normal force acts perpendicular to the rim surface, pushing the chain outward and contributing to the separation from the edge that forms the arched fountain. Frictional forces, though often small in smooth setups, act tangentially to oppose sliding and help manage the change in direction, effectively redirecting the chain's linear momentum into a curved path. These interactions prevent the chain from simply draping over the rim and instead impart the upward component essential to the phenomenon. Gravity exerts a uniform downward force on every segment of the chain, proportional to its mass per unit length, which drives the overall descent and energy release but challenges the rising portion of the fountain. This downward pull is balanced in the arched bend by a dynamic lift arising from the net effect of rim contact forces and tension, enabling the chain to rise above the rim despite gravitational opposition. Without this balance, the fountain would collapse into a simple siphon. Basic free-body diagrams illustrate these forces on representative chain elements:
  • Element in the pile (stationary bead being engaged): acts downward; low tension pulls upward from the adjacent moving link; a reaction from the stationary pile provides an upward push to initiate motion, arising from the pile's resisting the of the bead.
  • Element at the rim (bead in the bend): Incoming and outgoing tensions act along the chain directions; pulls downward; from the rim pushes radially outward; frictional opposes tangential motion.
  • Element in the fall (descending bead): High tension acts upward from the chain above; pulls downward, resulting in net ; air resistance is typically negligible.

Theoretical Explanations

Initial Hypotheses

Early explanations for the chain fountain phenomenon drew analogies to , particularly likening the chain's behavior to a self-siphoning liquid, where the falling portion pulls the stationary over the rim without external input. This analogy suggested that the acts like an incompressible fluid, with the of the descending links driving continuous flow. However, it failed to account for the characteristic arch height, as a simple would not produce the observed upward rise beyond the container's rim. Subsequent hypotheses invoked momentum conservation, positing that tension in the chain accelerates both stationary and moving segments equally, leading to an upward force on the emerging links. These simple models assumed the pull from the falling chain directly imparts to the coiled portion without additional influences. Their limitation lay in neglecting interactions at the container rim, such as frictional or impulsive forces, resulting in predictions of infinite at the takeoff point that contradicted the smooth, finite arch observed. Energy-based arguments proposed that gravitational released by the falling chain converts into for the fountain, sustaining the motion through efficient transfer. Proponents claimed minimal dissipation allows the structure to persist until the chain depletes. Yet, these ideas proved inconsistent with the phenomenon's longevity, as classical estimates predicted excessive energy loss (up to half the ) during link pickup, which would halt the fountain prematurely rather than maintain the steady arch.

Biggins and Warner Model

In 2014, John Biggins and proposed a theoretical model for the chain fountain phenomenon, attributing the upward motion to a reaction force arising from the change as chain segments accelerate from rest at the rim. This force, derived from the rate at which stationary elements are set into motion with speed vv, is given by F=12μv2,F = \frac{1}{2} \mu v^2, where μ\mu is the linear mass density of the . The derivation considers the impulse required to change the of each link from zero to μv\mu v per unit length, resulting in an effective upward that opposes the chain's fall and drives the fountain formation. Central to the model are the dynamics of , governed by its inherent , which resists sharp at the rim. As transitions from the stationary pile to the moving vertical segment, tension in generates a that must be balanced by the . This leads to a arched configuration, where forms a curved apex rather than a straight drop, with the determined by the equilibrium between tensile s and elastic resistance to . The arch shape minimizes while accommodating the momentum-driven , ensuring the fountain's persistence until the chain depletes. The model predicts that the height of the arch hh scales as h∝v2/gh \propto v^2 / g, where gg is , reflecting the balance between the inertial upward force and gravitational pull on the rising segment. This quadratic dependence on chain speed was validated through experiments using a 50-meter nickel-plated ball chain, where measured arch heights closely matched the theoretical proportions, with h≈0.14h1h \approx 0.14 h_1 ( h1h_1 being the initial pile height) for typical speeds. The framework assumes an with uniform and neglects frictional losses at the rim or within the pile, focusing on steady-state dynamics in a frictionless environment. These simplifications enable analytical tractability but limit applicability to real chains exhibiting dissipative effects or variable conformations.

Post-2014 Developments

Following the foundational model proposed by Biggins and Warner in 2014, subsequent theoretical work has sought to generalize and refine the understanding of the chain fountain effect through alternative proofs, enhanced analyses, and computational approaches. In , researchers demonstrated via a proof-by-construction that the chain fountain emerges as a generic outcome of energy-conserving linear dynamics in chains, independent of specific properties or container interactions beyond basic conservation laws. This approach constructs the phenomenon step-by-step from idealized linear chain segments under tension, showing how the self-siphoning arch forms solely from the interplay of kinetic and without invoking external impulses, thereby broadening the effect's applicability to various chain configurations. A 2021 theoretical revisit provided a refined of the siphoning mechanism, emphasizing the role of in achieving the necessary for the fountain while conserving both and during the pickup phase from the . This work modeled the as a flexible, inextensible and derived conditions under which the falling segment's speed exceeds the threshold for sustained arching, incorporating detailed tension to explain why the effect persists even in setups minimizing contact forces from the base. Numerical simulations have further advanced post-2014 insights by modeling non-ideal chains with finite and bead-like discreteness, predicting variations in arch and based on constraints and packing irregularities. For instance, discrete element simulations using bead-spring models with angular limits (e.g., maximum of 63°) reproduce observed fountain proportional to the square of the chain speed, while also showing how smooth container bottoms eliminate the effect, whereas bumpy bottoms enable it, attributing deviations to localized transfers at pickup points. These methods reveal that arch tightens with increased chain rigidity, offering quantitative predictions for experimental deviations from ideal smooth arches. Ongoing debates center on the relative contributions of chain versus drag in driving the fountain's rise. Proponents of emphasize its role in sustaining the arch against , as rigid resist and amplify upward forces, whereas drag advocates argue that inertial effects from accelerating segments dominate, with merely modulating secondary shape variations; simulations support a hybrid view where drag provides the input, but prevents through tangling.

Experimental Investigations

Terrestrial Experiments

Terrestrial experiments on the chain fountain have employed high-speed cameras to profile the and of the chain segments as they emerge from the container and form the arch. For instance, cameras operating at 1000 frames per second have captured the motion in detail, allowing for precise tracking of the chain's pickup point and arch height. Complementing this, sensors, such as PASCO force platforms placed under the container, measure the reaction exerted by the chain pile, providing data on the transfer during steady-state flow. These techniques confirm the v2v^2-dependent lift mechanism, where the upward scales with the square of the chain's , aligning with theoretical benchmarks for arch formation. Key findings from these ground-based studies demonstrate that the arch height increases quadratically with velocity, with measured heights matching predicted values; for example, an arch height of 15 cm was observed at a chain speed of 2 m/s using a standard cup-shaped container. In another setup with a crown-shaped pot, heights reached 25 cm at 4.2 m/s, highlighting the role of the pickup dynamics in sustaining the fountain under Earth's gravity. Force measurements under the container reveal a steady reaction force proportional to the chain's linear density and velocity squared, typically on the order of several newtons for bead chains falling from heights of 1-3 m, validating the momentum balance without significant deviations from predictions. Experiments have tested variations in mass, pot geometry, and ejection speeds to isolate contributing factors. Chains with linear densities around 21 g/m, composed of 2.8 mm beads, were compared to heavier variants like 4.5 mm beads at 50 g/m, showing taller arches for lower- chains at equivalent speeds due to reduced . Pot shapes ranged from simple cups to crowned rims and ashtrays with varying openings (e.g., 4.5-12.8 cm), where narrower openings increased tangling but enhanced the whip-like transfer. Speeds from 2 to 6 m/s were achieved by adjusting drop heights of 1.3-3.6 m, revealing that higher velocities amplify the lift but are limited by length and piling method. Error sources in these experiments include chain tangling, which delays the and reduces effective height by up to 20% in coiled configurations, quantified through video analysis of unfolding phases. Air resistance contributes minimally at speeds above 2 m/s, adding less than 5% for typical bead diameters, while at the rim—estimated from fluctuations—accounts for 10-15% dissipation, particularly in bar-link chains versus spherical . These effects were mitigated by using smooth rims and wall-piling arrangements, ensuring reproducible results under standard gravitational conditions.

Microgravity Tests

In 2025, astronaut Don Pettit performed a microgravity experiment on the (ISS) to investigate the chain fountain effect without gravitational influence. The setup consisted of a 4-meter bead arranged in a closed loop and stretched into a rectangular configuration to mimic the self-sustaining flow observed terrestrially, with the process guided remotely by physicist and filmed for analysis. Pettit initiated the motion by pulling on the , allowing it to circulate freely in the zero-gravity environment of the ISS. The observations demonstrated persistence of the chain's self-flowing behavior, with the chain circulating through its own loop and retaining an angular shape initially, though it gradually rounded due to interactions with air currents and . Unlike Earth-based demonstrations, where induces a falling arch, the zero-g conditions eliminated downward , resulting in an altered, floating that lacked the typical siphoning descent but still exhibited sustained motion without external support. These findings imply that the chain fountain arises primarily from momentum-based interactions rather than relying on a pure gravitational siphon, as the effect endured solely due to the chain's speed vv. Video footage analysis indicated a reduced yet evident lift force propelling the chain, consistent with modified theoretical models for microgravity, and provided a comparative baseline to terrestrial results where gravity enhances the arch's height and stability.

Variations and Extensions

Alternative Configurations

Alternative configurations of the chain fountain involve modifications to the chain structure or experimental setup, which can alter the persistence and of the arch while preserving the core in Newtonian contexts. chains, consisting of spherical elements connected by short rigid rods, exhibit high flexibility with a maximum bending approaching 180°, allowing the chain to conform closely to the container's surface but requiring a bumpy bottom to initiate and sustain the upward deflection through transfer during link rotations. In contrast, link chains, such as those with interconnected rigid segments limiting the bending to approximately 63°, provide greater structural rigidity that enhances arch persistence by resisting collapse and supporting higher fountain elevations, as the constrained flexibility amplifies the upward from the container's edge. For instance, bar-link chains with extended cylindrical segments demonstrate a higher rise-to-fall of 0.14 compared to 0.12 for standard chains, due to their reduced tangling and improved transmission. Horizontal setups, such as rimless configurations where the chain is picked up directly from a flat table without a raised edge, isolate the deflection forces by eliminating rim interactions and confirm that the upward push originates from the supporting surface during extraction. In these arrangements, a horizontal segment is drawn away while the remaining pile provides the reactive force, producing a lateral "" arc analogous to the vertical case but without gravitational dominance over the initial motion. Inverted or multi-pot variants, though less common, extend this by distributing the across multiple containers to test sustained flow, revealing that the effect persists as long as surface interactions generate the necessary kick, albeit with reduced arch height in segmented setups. Scaled versions highlight size-dependent behaviors, where micro-scale chains—such as those with 3 mm beads—exhibit more pronounced tangling and lower rise heights due to increased relative , while long industrial-scale chains, up to 50 m in , maintain stable fountains with arches scaling quadratically in time before reaching , as longer lengths amplify cumulative . For example, small-bead chains ( 0.30 cm) achieve consistent rise ratios independent of fall , but larger bar chains ( 1.02 cm) show enhanced stability at greater scales. The chain fountain effect extends to analogies in other materials, such as , where a continuous flexible coiled in a produces a similar self-sustaining arch upon extraction, driven by tension rather than discrete links, though with diminished height due to the absence of rigid segments. In granular materials, chains behave analogously by forming force networks akin to particle interactions, exhibiting fluid-like flow during extraction and solid-like resistance in the pile, which parallels the transfer in formation.

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