Impacting spheres: from liquid drops to elastic beads

Abstract

A liquid drop impacting a non-wetting rigid substrate spreads laterally, then retracts, and finally jumps off again. An elastic solid, by contrast, undergoes a slight deformation, contacts briefly, and bounces. The impact force on the substrate – crucial for engineering and natural processes – is classically described by Wagner's (liquids) and Hertz's (solids) theories. This work bridges these limits by considering a generic viscoelastic medium. Using direct numerical simulations, we study a viscoelastic sphere impacting a rigid, non-contacting surface and quantify how the elasticity number (El, dimensionless elastic modulus) and the Weissenberg number (Wi, dimensionless relaxation time) dictate the impact force. We recover the Newtonian liquid response as either El → 0 or Wi → 0, and obtain elastic-solid behavior in the limit Wi → ∞ and El ≫ 1. In this elastic-memory limit, three regimes emerge – capillary-dominated, Wagner scaling, and Hertz scaling – with a smooth transition from the Wagner to the Hertz regime. Sweeping Wi from 0 to ∞ reveals a continuous shift from materials with no memory to materials with permanent memory of deformation, providing an alternate, controlled route from liquid drops to elastic beads. The study unifies liquid and solid impact processes and offers a general framework for the liquid-to-elastic transition relevant across systems and applications.

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1 Introduction

Impacts of spherical bodies on rigid substrates span two classical limits that have long been treated separately: liquid drops^1–5 and elastic solids.^6–9 Both occur widely in nature and technology, where the normal force on the substrate is often the quantity of interest because it can damage engineered surfaces.^10,11 Drop impact governs fields ranging from inkjet printing¹² and spray cooling/coating^13,14 to forensics,¹⁵ pesticide deposition,¹⁶ and soil erosion.¹⁷ Impacts of elastic solids arise in hardness testing,¹⁸ granular media and suspensions,¹⁹ sports,²⁰ and everyday bouncing of soft rubber balls. Despite this breadth, a unifying framework for the impact force across liquid and solid limits has remained elusive.

A falling liquid drop, after impact on a rigid surface, deforms and spreads laterally until it reaches its maximum extent. A pronounced peak in the temporal evolution of force occurs at the instance of drop touchdown on the surface due to the inertia of the impact, whereas during droplet spreading this force is much smaller.⁵ For perfectly wetting surfaces, the liquid sticks to them. However, for non-wetting surfaces, the drop retracts from its maximum spread and generates a Worthington jet which coincides with a second peak in the temporal evolution of force.^3,4 In the inertial regime, Wagner's theory predicts that the impact force scales as

F ∼ ρ_lV₀²R₀², (1)

where ρ_l is the density of the media, V₀ is the impact velocity and R₀ is the radius of the falling drop.¹¹

By contrast, the elastic solids undergo slight deformation on impact with the substrate and bounce off following a brief contact with the substrate due to the exerted normal reaction. In such cases, the temporal evolution of force is characterized by a single maximum. Assuming the contact area to be small in comparison to the bead's size, and considering a non-adhesive contact with small strains within the elastic limit, the situation can be treated as a Hertzian contact problem.⁹ Thus, Hertz's theory describes the scaling laws for the impact force in the case of the impact of an elastic bead on a rigid substrate,

F ∼ (GR₀²)^2/5(ρV₀²R₀²)^3/5 (2)

where G is the modulus of rigidity, ρ is the density of the solid medium, and R₀ and V₀ as mentioned before are the radius and impact velocity of the solid elastic bead.

Viscoelastic media – here, soft elastic gels – bridge liquids and solids: when deformed they support both viscous flow and recoverable elastic stress.²¹ Compared with Newtonian liquids, their rate-dependent rheology can markedly alter spreading, pinch-off, and rebound on impact.^22,23 Such soft media are relevant to inkjet printing,²⁴ drop deposition,²⁵ and spray atomization.²⁶ Soft solids such as hydrogels, comprising cross-linked networks with tunable elasticity, are widely used as biocompatible materials in rapid prototyping²⁷ and drug delivery;²⁸ see Chen et al.²⁹ for background.

In this work, we parameterize the gel's elastic response using the elastic modulus G which is the proportionality constant between strain and elastic stresses, and the relaxation time λ that sets the decay timescale of those elastic stresses. Upon non-dimensionalizing the governing equations (Section 2.1), two material control parameters emerge. The elastocapillary number

compares the elastic modulus to the Laplace pressure, while the Deborah numbercompares the elastic-stress relaxation timescale to the inertio-capillary process time. Here γ represents the coefficient of surface tension.

The impact inertia is expressed by the Weber number

which compares the inertial and capillary forces. Another important dimensionless control parameter of the system is the Ohnesorge numberwhich is the ratio of inertio-capillary and the visco-capillary timescales. Here η_l is the dynamic viscosity of the sphere.

Two combinations of these numbers will be central in what follows. The elasticity number

compares elastic to inertial stresses. The Weissenberg numbercompares the elastic relaxation time λ to the impact time R₀/V₀ (see Section 2.1).

In this study, we simulate impacts of soft gel spheres on a rigid, non-contacting substrate using a volume-of-fluid, finite-volume framework. By varying the elastocapillary number (Ec) and the Deborah number (De), we traverse smoothly from liquid-like to solid-like response and compare the resulting force scalings with Wagner's (liquids) and Hertz's (elastic solids) theories. We develop an expression for the peak force that transcends the two regimes, using a function of the elasticity parameter to compare the shear modulus with the impact stress. Consistent with these limits, we recover Newtonian-liquid behavior for De = 0 or Ec = 0, while for De → ∞ at sufficiently large Ec the dynamics converge to those of an elastic solid.

2 Numerical framework

2.1 Problem description and governing equations

We consider an axisymmetric sphere of radius R₀ approaching a rigid substrate with initial velocity V₀. For liquid drops the substrate is non-wetting; for viscoelastic and elastic beads it is non-contacting. The sphere is a viscoelastic medium of density ρ_l, dynamic viscosity η_l, elastic modulus G, relaxation time λ, and surface tension coefficient γ. The surrounding gas has density ρ_g and viscosity η_g (Fig. 1).

Fig. 1 Schematic: a viscoelastic sphere (radius R₀) impacts a non-contacting rigid surface with velocity V₀. Material properties are ρ_l, η_l, G, λ, and γ for the sphere; ρ_g and η_g for the gas.

Lengths are scaled by R₀ and time by the inertio-capillary timescale . The corresponding velocity and pressure/stress scales are and σ_γ = γ/R₀, respectively. Throughout the manuscript, all variables with a tilde are non-dimensionalized using the above mentioned scales. The incompressible mass and momentum balances in the viscoelastic phase read

∇·u = 0 (9)

andwhere f_γ is the non-dimensional capillary force density acting at the interface. The Newtonian (viscous) stress iswith denoting the symmetric part of the velocity gradient tensor. Furthermore, the normal force on the substrate is obtained using the rate of change of sphere's momentumwhere V_cm denotes the velocity of the centre of mass of the drop at any instant.

The elastic stresses arise from deformation of the microstructure quantified by the conformation tensor .²¹ Using the Oldroyd-B constitutive model,³⁰

with the elastocapillary number Ec = GR₀/γ (eqn (3)). The conformation tensor relaxes to on the Deborah timescale De = λ/τ_γ (eqn (4)) viawhere the upper-convected derivative is

The Deborah number De quantifies material memory: De = 0 recovers a Newtonian liquid characterized by Oh; De → ∞ yields an elastic solid limit with

and Oldroyd-B equivalent to a neo-Hookean solid.²¹

Despite being widespread due to simplicity, the Oldroyd-B model suffers from certain limitations.^21,31 It fails to capture the shear-thinning behaviour in viscoelastic fluids completely, and erroneously predicts unbounded stress growth in strong extensional flows.³² This limitation can be addressed by incorporating the finite polymer extension (FENE-P model).³³ Also various other extensions of the Oldroyd-B model have been developed³⁴ like the Phan–Thien–Tanner (PTT)³⁵ model to account for such non-linearities. However, since we are not dealing with strong extensional flows, we restrict ourselves to the Oldroyd-B model in this study, as it is sufficient to describe our case.

2.2 Numerical methods and simulations

We solve the above equations with Basilisk C,³⁶ using a one-fluid formulation with surface tension as a singular interfacial force.^37,38 The liquid–gas interface is tracked by a volume-of-fluid (VoF) color function ψ advected bywith ψ = 1 in liquid, ψ = 0 in gas, and 0 < ψ < 1 at the interface. Mixture properties are

ρ = ψρ_l + (1 − ψ)ρ_g, (18)

η = ψη_l + (1 − ψ)η_g, (19)

with fixed ratios ρ_r = ρ_g/ρ_l = 10⁻³ and η_r = η_g/η_l = 10⁻² and a small sphere Ohnesorge number, Oh = 10⁻². A geometric VoF reconstruction applies capillary forces as

f_γ ≈ γκ∇ψ, (20)

where the curvature κ is computed via height functions.³⁹ Explicit surface-tension forcing imposes the standard capillary time-step constraint;⁴⁰ the explicit update of σ_e adds a typically milder constraint.

At the substrate we impose no-penetration and no-slip, and a zero normal pressure gradient. To enforce a non-contacting (superhydrophobic) condition we set ψ = 0 at the wall, maintaining a thin air cushion.^41,42 For liquid drops, enforcing this air cushion results in a non-wetting (superhydrophobic) substrate, while for elastic spheres this results in a non-contacting substrate. We stress that contact initiation in soft-solid impacts is generically air-mediated and can proceed annularly or patchily with a non-monotonic initial contact radius; see Zheng et al. (2021)⁴³ for direct observations of air-mediated contact in compliant-hemisphere impacts. Top and lateral boundaries use outflow (ambient pressure, zero tangential stress, and zero normal velocity gradient). Boundaries are positioned far enough to avoid spurious confinement effects. The axisymmetric domain size is 8R₀ × 8R₀. We employ quadtree adaptive mesh refinement (AMR)^40,44 with maximal refinement at the interface and in regions of large velocity gradients. Wavelet-based error control uses tolerances 10⁻³ for u, ψ, κ, and . Grid-independence tests confirm convergence. Unless stated otherwise, the minimum cell size is Δ = R₀/512 (i.e. 512 cells per radius on a uniform equivalent grid), increased to Δ = R₀/2048 when required. Further numerical details can be found in studies of Sanjay,⁴² Popinet⁴⁴ and Dixit et al.³¹

3 Wagner versus Hertz: permanent-memory impacts

In this section, we quantify the solid-impact limit by taking De → ∞, so the material retains its deformation memory over the process time. In the numerics, we keep a small background viscosity, so the spheres are Kelvin–Voigt solids rather than perfectly elastic; this facilitates comparison with inertial liquid impacts at finite Oh and avoids the numerical breakdown of the inviscid (Euler)–elastic limit. We therefore approach the purely elastic response by letting Oh → 0.

We sweep the (Ec,We) space over We ∈ [1, 10³] and Ec ∈ [10⁻¹, 10⁴]. The normal reaction on the substrate is computed from the drop's momentum balance (eqn (12)). For liquid drops on non-wetting substrates, F(t) exhibits two peaks: an inertial peak at touchdown and a later peak associated with the formation of a Worthington jet.³ The second, jetting peak occurs at a time t₂ ≫ t_max, with the ratio ,⁴ reflecting the phase difference between recoil and jet formation. In contrast, for elastic spheres the loading and unloading remain nearly in phase: the entire contact–rebound cycle fits within t ≲ 2t_max and F(t) displays only a single, almost symmetric peak in this interval. To compare liquids and solids and to track the transition, we therefore focus on the first (inertial) peak F_max, non-dimensionalized as F_max/(ρ_lV₀²R₀²); the corresponding time is t_max.

Fig. 2 shows representative cases across the parameter space. At low Ec the sphere flows and behaves liquid-like (Fig. 2b and c; Ec = 1, 2). At low Ec and low We, capillarity is significant (Fig. 2b). At high Ec the sphere deforms slightly and rebounds after a short contact (Fig. 2d; Ec = 1000). Increasing We at fixed Ec effectively softens the response (reduces El) and increases the contact duration (Fig. 2d). The force traces reflect this evolution: for large El (high Ec, low We) F(t) is nearly symmetric, as in elastic impacts, while decreasing El (e.g. by increasing We) skews F(t) in the manner typical of liquid impacts.^3,4 The peak magnitude also varies appreciably across cases. Since force and time are related as [F_max/(ρ_lV₀²R₀²)]·[t_max/(R₀/V₀)] ∼ 1 for any impacting sphere in general, t_max reduces accordingly with an increase in F_max.

Fig. 2 (a) Phase space in the Ec–We plane illustrating the range of simulations conducted in this work colored according to the elasticity number El = Ec/We. The four highlighted symbols locate typical cases representing the range of parameters explored. We chose (We,Ec) = (b) (5, 1), (c) (500, 2), (d) (5, 1000), (e) (500, 1000). In each case, the color scheme of each snapshot represents the magnitude of the velocity normalized by the impact velocity, alongside the corresponding force history F(t)/(ρ_lV₀²R₀²) plotted versus t/t_max (right). The force traces are plotted up to t/t_max = 2: for the liquid-drop reference, the second peak associated with the Worthington jet^3,4 occurs at later times t₂ ≫ t_max (with ) and is therefore outside the plotted window.

The dependence of F_max on We and Ec is summarized in Fig. 3. For Ec ≲ 1, F_max follows the liquid-impact trend with the low-We correction,³

Fig. 3 Peak force in the elastic-memory limit (De → ∞): (a) Variation of the normalized peak force, F_max/(ρ_lV₀²R₀²), with the Weber number We = ρ_lV₀²R₀/γ for different elastocapillary numbers Ec = GR₀/γ. For the data follow the liquid-impact result: a high-We Wagner plateau ≃3.24, with the low-We correction F_max/(ρ_lV₀²R₀²) ≈ 3.2/We + 3.24 (dashed line, eqn (21)). As Ec increases, F_max rises, most clearly at low We, and for sufficiently large Ec the curves acquire a log–log slope −2/5, i.e. F_max/(ρ_lV₀²R₀²) ∼ We^−2/5 at fixed Ec, consistent with the approach to Hertzian elastic contact. (b) Dependence on Ec at fixed We (curves labelled by We). At small Ec all series collapse to the liquid-like level (≈3.24); above a We-dependent crossover, F_max increases monotonically with Ec, following F_max/(ρ_lV₀²R₀²) ∼ Ec^2/5, again, consistent with the approach to Hertzian elastic contact. Together, (a) and (b) show a continuous evolution from Wagner (liquid) to Hertz (elastic) behavior as Ec increases.

As Ec increases, F_max rises, most strongly at low We. At sufficiently large Ec, F_max/(ρ_lV₀²R₀²) decreases with We with a log–log slope ≃ −2/5, indicating a transition from eqn (21) to F_max/(ρ_lV₀²R₀²) ∼ We^−2/5. At fixed We (Fig. 3b), F_max is nearly constant at small Ec and then increases steadily with Ec, with higher magnitudes at lower We.

Collapsing the data using El (Fig. 4a) reveals two regimes. For El ≲ 1 and sufficiently large We, the data lie near Wagner's constant level, F_max/(ρ_lV₀²R₀²) ≈ 3.24. At small We, inertia competes with capillarity and the low-We correction in eqn (21) is required.⁴² For El ≳ 1, all points collapse onto a single master curve with slope 2/5:

consistent with Hertz scaling for elastic impacts. Notably, surface tension does not enter this high-El law, as expected for elastic solids. The transition from the Wagner (liquid) to the Hertz (elastic) regime is smooth. A contour map over (We, Ec) (Fig. 4b) visualizes the continuous variation of F_max across the space.

Fig. 4 Unified scaling and regime map: (a) collapse of the normalized peak force versus the elasticity number El = Ec/We = G/(ρ_lV₀²). For El ≲ 1 and sufficiently large We the data sit on the Wagner plateau F_max/(ρ_lV₀²R₀²) ≈ 3.24; deviations at very small We reflect capillary corrections in eqn (21). For El ≳ 1 all cases collapse onto a single power law with slope 2/5, F_max/(ρ_lV₀²R₀²) ∼ El^2/5 (eqn (22)), the hallmark of Hertz scaling. (b) Contours of F_max/(ρ_lV₀²R₀²) in the (We,Ec) plane (symbols: simulation points). The dashed guide El = 1 marks the smooth crossover from the Wagner region (lower right) to the Hertz region (upper left); the low-We corner is capillary-dominated and requires the correction in eqn (21). The map visualizes the continuous transition from liquid-like to solid-like impact forces as We and Ec are varied.

4 Theory

In this section we identify the asymptotic limits of the peak impact force and develop a unified predictive model. First, we derive scaling expressions for F_max in the two extreme regimes – an elastic contact limit versus a hydrodynamic impact limit – and then combine these results to propose a single predictive expression for the dimensionless maximum impact force.

4.1 Purely elastic limit: Hertz contact theory

Consider a solid elastic sphere of radius R₀, mass m, and elastic modulus G (shear modulus, assuming an incompressible material) impacting a rigid flat surface with speed V₀. Upon contact, the sphere deforms and a normal force F develops according to Hertz's contact law. For a sphere indenting a half-space, the force–indentation relation is given by the 3/2-power law of classical Hertz contact mechanics,⁶where δ(t) is the indentation depth and E* is the effective Young's modulus of the contacting pair. For a sphere against a rigid flat, E* = 2G/(1 − ν); taking Poisson's ratio ν ≈ 0.5 for an incompressible solid, we get E* ≈ 4G, so the prefactor in eqn (23) is about .

In the ideal elastic limit (no dissipation), the sphere will momentarily come to rest at maximum compression, converting all its kinetic energy into elastic deformation energy. Using energy conservation between the moment of impact and the instant of maximum indentation δ_max (when ), we have:

Substituting the Hertz law for F(δ) and performing the integration yields the elastic energy stored at indentation δ_max:

Rearranging this result to solve for the peak indentation δ_max gives:

The maximum force occurs at δ = δ_max. Substituting the expression for δ_max in eqn (23), we get

Normalizing with the inertial force scale ρ_lV₀²R₀², we get

which is consistent with our large El results, cf. Fig. 4a.

4.2 Purely liquid limit: Wagner impact theory

At the opposite extreme (Ec → 0) the sphere behaves as a liquid drop, and its impact dynamics are governed by inertia and capillarity in the classical Wagner limit.^3,11 Instead of an elastic compression, the drop undergoes rapid localized deformation at the moment of impact: the south pole flattens against the substrate while the remainder of the drop (including the north pole) is still moving downward at nearly the impact speed. The vertical momentum of the drop's center of mass is redirected into a radial outflow along the substrate, causing a small “wetted” area to grow outward from the impact point.⁴ This Wagner-type mechanism – a thin spreading lamella initiated at the contact point^45–48 – contrasts sharply with the distributed Hertzian contact of an elastic solid. It produces a pronounced impulsive force at touchdown, as the drop's momentum is arrested over a short time and small area. The first force peak thus originates from pure inertial impingement of the liquid on the surface.⁵ We have analyzed this case in past; for details see the references Zhang et al.,³ Sanjay et al.,⁴ Sanjay and Lohse.⁵

During the very early stage (t ∼ τ_ρ = D₀/V₀), the normal force rises sharply to its first maximum F_max as the drop's inertia is transferred to the substrate. At this moment the deformation is still localized: the contact radius has grown only to the order of the drop's initial radius. In fact, experiments confirm that at the peak force time t_max, the spread diameter D_f(t_max) is approximately equal to the initial drop diameter D₀, consistent with early-time self-similarity of the impact.^4,5,49,50 Wagner's inviscid theory predicts that the peak force scales with the inertial pressure on the drop's footprint. Non-dimensionalizing F_max by ρ_lV₀²R₀² (with ρ_l the liquid density) yields a constant of order unity. Indeed, for large Weber numbers (negligible surface tension), simulations and experiments find F_max/(ρ_lV₀²R₀²) ≈ 3.24, cf. Fig. 3a of Zhang et al.³ At lower We, surface tension enhances the impact, and the peak force increases with decreasing We (following a F_max/(ρ_lV₀²R₀²) ∼ We⁻¹ correction in this regime). This initial peak is inertia-dominated and is relatively insensitive to liquid viscosity: F_max remains nearly constant for drops with viscosity up to about 100× that of water.⁴ Since our spheres have a very low background viscosity (Oh = 10⁻²), in the limit Ec → 0 our results for F_max follow the same trends. Only for highly viscous drops (Ohnesorge number Oh ≳ 1) does viscous dissipation significantly attenuate the first peak, reflecting the fact that most of the drop's momentum is redirected (and the force generated) before substantial viscous effects have time to act, cf. Sanjay and Lohse.⁵

4.3 Predictive interpolating model for maximum impact force

For intermediate conditions , the sphere's deformation and the fluid's inertia both contribute, and the peak force deviates from either pure Hertz or Wagner scaling alone. We therefore express the dimensionless peak force as a weighted transition between the two asymptotic contributions following the approach of Sanjay & Lohse (2025),⁵with a smooth transition function f(El) based on tanh function defined aswhere the We-dependent parameters a ≈ 1.3 for We → 1 and ≈ 0.5 for We ≳ 10, and b ≈ 1.4 for We → 1 and ≈ 0.2 for We ≳ 10 (see ref. 51 for details of this fit). Here, a measures the critical El at transition from Wagner's to Hertz's scaling (Fig. 4). The width of this transition is indicated by b. The remaining coefficients in eqn (31) are fixed a priori: the Hertz prefactor 5.3 follows directly from the elastic analysis above (without fitting), while the constants 3.24 (Wagner plateau) and 3.2 (the We⁻¹ capillary correction) are taken from the Newtonian impact model of Sanjay & Lohse (2025).⁵ This construction ensures a continuous interpolation between the Hertz and Wagner limits. In spirit it follows the additive scaling approach of Sanjay & Lohse (2025)⁵ for drop impacts, but unlike their model – which includes separate viscous regimes – here only the two primary regimes (elastic vs. inertial) are needed. The resulting formula smoothly bridges the two asymptotes and correctly reproduces the peak-force scaling in both limits (this predictive curve is plotted in Fig. 4a for comparison).

5 Influence of elastic stress relaxation

The results in Section 3 established the two asymptotic force scalings for impacts with permanent memory (De → ∞): a Wagner plateau at a small elasticity number and a Hertz law at a large elasticity number (Fig. 4). These two limits also bound impacts when the material memory is finite: relaxing the memory shifts the response continuously from solid-like to liquid-like, with the transition controlled by the non-dimensional relaxation time. We quantify the memory effect with the Weissenberg number, Wi (eqn (8)), which compares the elastic relaxation time λ to the impact time R₀/V₀ (see Section 2.1). While De compares λ to the inertio-capillary time, Wi is the more natural process-time measure here and increases with the degree to which elastic stresses persist during impact.⁵² Thus Wi = 0 (De = 0) recovers a Newtonian liquid, whereas Wi → ∞ (De → ∞) yields an elastic-memory limit.

Fig. 5 (fixed We = 100 and El = 40) visualizes the progressive loss of elastic behavior as Wi decreases from Wi → ∞ to 0. In the large-Wi limit, the bead contacts briefly and rebounds; the force trace is nearly symmetric with a large peak, characteristic of Hertz-like loading. Reducing Wi increases the contact time and skews F(t) towards a liquid-like evolution with a much smaller peak. At Wi = 0 the material has no memory and behaves as a Newtonian liquid: the sphere spreads and flows, and F(t) exhibits the familiar asymmetric shape.

Fig. 5 Relaxing material memory at fixed We and El. Evolution of shape (left) and force (right) when Wi decreases from ∞ to 0 at We = 100 and El = 40: (a) Wi → ∞ (elastic-memory limit); (b) Wi = 10⁻¹; (c) Wi = 10⁻²; (d) Wi = 0 (Newtonian). For each case, the color scheme of each snapshot represents the magnitude of the velocity normalized by the impact velocity, alongside the corresponding force history F(t)/(ρ_lV₀²R₀²) plotted versus t/t_max (right). As Wi decreases, the contact time increases and F(t) becomes increasingly liquid-like with a reduced F_max.

To quantify the role of memory, we plot F_max/(ρ_lV₀²R₀²) versus El = Ec/We = G/(ρ_lV₀²) for several Wi values at We = 100 in Fig. 6. At Wi = 0, the data follow the Newtonian liquid level established in Section 3. On the other hand, as Wi → ∞, the points converge to the elastic-memory master curve of Section 3, namely the Hertz scaling F_max/(ρ_lV₀²R₀²) ≈ 5.3El^2/5 at large El. Between these limits, weaker memory (smaller Wi) delays the transition to Hertz scaling, while stronger memory (larger Wi) makes the elastic response apparent already for softer spheres. Notably, even at Wi → ∞ the plateau persists for El ≪ 1, because the modulus is introduced exclusively through El: vanishing G implies a liquid-like bound irrespective of memory.

Fig. 6 Peak force versus elasticity number at different Wi (all at We = 100). The black horizontal line indicates the Wagner plateau (≃3.24); the dashed guide has slope 2/5 (Hertz). Increasing Wi shifts the departure from the plateau to lower El and drives the curves toward the Hertz master law; at Wi = 0 all data sit on the Wagner level.

Taken together with Section 3, these results show that both material stiffness (via El) and material memory (via Wi) govern the peak impact force: the Wagner and Hertz laws remain the bounding asymptotes, while Wi sets how rapidly the system transitions between them.

Conclusions and outlook

In this work, we investigate the impact of a viscoelastic sphere on a non-contacting rigid surface and chart a continuous transition in impact dynamics from liquid-like to solid-like behavior by tuning the material parameters. The primary peak force F_max (associated with the inertial impact) smoothly crosses over from Wagner's inertial-drop scaling to Hertz's elastic-contact scaling as the elasticity number El = Ec/We increases. For small El (liquid-like response), we reproduce F_max/(ρ_lV₀²R₀²) → 3.24 matching the constant plateau from Wagner's theory. In contrast, for large El (elastic-dominated regime), F_max/(ρ_lV₀²R₀²) grows following a power-law ≈5.3El^2/5, consistent with Hertz's prediction for elastic spheres. These two limiting behaviors bound the force response, and the transition between them is gradual rather than abrupt. The Weissenberg number , which quantifies the polymer's relaxation time relative to the impact time, governs this memory-driven crossover: as Wi increases from 0 (no elastic memory) to ∞ (permanent memory), the peak-force scaling shifts continuously from the Wagner limit to the Hertz limit. Thus, by adjusting Wi, one can smoothly interpolate between liquid-drop and elastic-solid impact outcomes.

The modeling choices in this work were made to isolate the physics of the liquid-to-elastic transition and enable direct comparison to the classical limits. The viscoelastic sphere obeys Oldroyd-B, which neglects finite microstructure extensibility and shear-thinning; this is acceptable for our flow history but can be systematically relaxed with constitutive models that incorporate finite extensibility and rate-dependent viscosity.^21,32,34,35 The substrate is non-contacting, so the thin gas layer is present but not explicitly resolved with lubrication and wetting dynamics; prior work shows that air cushioning, skating on a gas film, and nanoscale first contact depend sensitively on slip, compressibility and rarefaction.^53–64 We stress that in the present work, the substrate is assumed to remain perfectly rigid. Small elastic deformations of the substrate can, however, modify the intervening gas layer,⁶⁵ which in turn may alter the impact dynamics and the measured force signature of the impacting sphere. Furthermore, the onset of contact will modify the shear stress at the interface – including effects such as adhesion – that can modify the peak force, and will alter the stresses upon rebound. Finally, a small background viscosity renders the (De → ∞) limit to be of the Kelvin–Voigt type rather than being ideally elastic; classical analyses quantify how viscoelastic dissipation and elastic waves perturb the Hertzian impact.^7,8 These assumptions do not alter the governing exponents (Wagner versus Hertz) but they may affect quantitative prefactors and very-short-time, near-contact details.

Several direct extensions can sharpen and generalize these results. Experiments with soft hydrogel/elastomer beads at moderate V₀ can probe the crossover regime and test the full F(t) waveform (peak magnitude, rise time, and symmetry/skewness), leveraging recent studies on soft-solid and gel impacts and elastohydrodynamic bouncing,^9,66–71 and using established force-measurement protocols from liquid-drop impacts.^3,4,72–74 Furthermore, incorporating a thin-gas lubrication model with dynamic wetting (including slip and, if needed, rarefaction/compressibility) will resolve when and how contact initiates and how this feeds back on the very-early-time force.^{43,49,53,56–61,64} Linking impact memory with lubricated-impact and wetting transitions will place the near-contact force history on firmer ground.^75–79

Author contributions

V. S., J. K., and D. L. conceived the study. S. J. and V. S. planned the numerical simulations. S. J. performed the simulations and analyzed the data. V. S. and D. L. developed the theory. S. J. and V. S. designed the structure of the manuscript. S. J., V. S., D. L. and J. K. wrote the manuscript. V. S. and D. L. supervised the project. All authors discussed the results and approved the final manuscript.

Conflicts of interest

There is no conflict of interest.

Data availability

All codes used in this work are available as an open-source repository at https://github.com/comphy-lab/Soft-Sphere-Impacts.

Supplementary information (SI) contains simulation videos clearly demonstrating the striking contrast between liquid drop impact and elastic sphere impact. See DOI: https://doi.org/10.1039/d5sm01078k.

Acknowledgements

We would like to thank Vincent Bertin, Pierre Chantelot, Maziyar Jalaal, Andrea Prosperetti, and Jacco Snoeijer for discussions. This work was carried out on the national e-infrastructure of SURFsara, a subsidiary of SURF cooperation, the collaborative ICT organization for Dutch education and research. This work was sponsored by NWO – Domain Science for the use of supercomputer facilities. This work was supported by NWO-Canon grant FIP-II grant. V. S. acknowledges start-up funding from Durham University. Open Access funding provided by the Max Planck Society.

Notes and references

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3. B. Zhang, V. Sanjay, S. Shi, Y. Zhao, C. Lv and D. Lohse, Impact forces of water drops falling on superhydrophobic surfaces, Phys. Rev. Lett., 2022, 129, 104501.
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