Adhesion Number in Multiscale Interactions

• Adhesion number is a dimensionless ratio that compares adhesive forces to competing scales such as inertia, capillary energy, or elastic compliance.
• It organizes system behavior by identifying when adhesion dominates, enabling a reduced-order description in contexts like particle packings and bubble coalescence.
• The parameter adapts to different systems by incorporating measures like work of adhesion, surface energy release, and effective interfacial strength to characterize transitions.

Searching arXiv for recent and foundational papers on adhesion number and related adhesion metrics. Adhesion number denotes a class of quantities used to compare adhesive interactions with a competing physical scale such as inertia, capillary energy release, elastic compliance, or dissipative loss. The term does not have a single universal definition across the adhesion literature. In some subfields it is a formally defined dimensionless parameter, as in adhesive particulate packings and wall-attached bubble coalescence; in others, closely related roles are played by a scale-dependent Tabor number, a normalized force, a work of adhesion, or an experimentally extracted interfacial strength. The common purpose is to identify when adhesion is negligible, when it controls morphology or dynamics, and how multiple control variables collapse onto a reduced description (Liu et al., 2014, Demirkır et al., 9 Jan 2025, Persson et al., 2014).

1. Terminological scope and general forms

Across current usage, “adhesion number” refers less to a single canonical symbol than to a recurrent modeling strategy: construct a ratio in which adhesion appears in the numerator and a competing energetic, mechanical, or geometric scale appears in the denominator. This yields either a nondimensional control parameter or, in studies that do not introduce a named number, an adhesion-related metric that plays the same classificatory role.

Context	Adhesion quantity	Function
Adhesive particle packings	$Ad$	compares adhesion with particle inertia
Bubble coalescence on walls	$W_{a,\text{tot}}^*$	compares adhesion energy with released surface energy
Rough elastic contact	$\mu_{\mathrm T}(\zeta)$	classifies DMT-like versus JKR-like behavior by scale
Other interfacial systems	$\Gamma$, $\tau_C$, $W_{\mathrm{ad}}$, $F/(\sigma d)$	experimentally grounded adhesion descriptors

In adhesive small-particle packings, the dimensionless adhesion parameter is written as

$Ad=\frac{w}{2\rho_p U_0^2 r_p},$

or, in a closely related formulation,

$Ad=\frac{\omega}{2\rho_p U_0^2 R},$

with $w=2\gamma$, $W_{a,\text{tot}}^*$0, and for equal spheres $W_{a,\text{tot}}^*$1. In both forms, the interpretation is the same: adhesion is compared with an inertial impact scale set by density, size, and deposition velocity (Liu et al., 2014, Liu et al., 2015).

A distinct energy-ratio form appears in bubble coalescence on solid walls, where the central parameter is

$W_{a,\text{tot}}^*$2

Here the denominator is the capillary surface energy released by coalescence, and the numerator is an adhesion-energy estimate associated with the contact patches on the wall (Demirkır et al., 9 Jan 2025).

In rough-surface contact, the analogous classifier is the magnification-dependent Tabor number,

$W_{a,\text{tot}}^*$3

which determines whether adhesion at a given length scale is better described as DMT-like or JKR-like (Persson et al., 2014).

2. Adhesion number in adhesive particulate packings

The most explicit and influential use of the term occurs in the packing of dry micrometer-scale particles. There, $W_{a,\text{tot}}^*$4 unifies the effects of particle size, particle velocity, and work of adhesion into a single control parameter. Larger $W_{a,\text{tot}}^*$5 arises from stronger adhesion, smaller particles, or smaller impact speeds; smaller $W_{a,\text{tot}}^*$6 corresponds to inertia-dominated deposition (Liu et al., 2014).

This parameter organizes the structure of the packing. One simulation study identified a threshold-like transition around $W_{a,\text{tot}}^*$7. For $W_{a,\text{tot}}^*$8, packing fractions are scattered between approximately $W_{a,\text{tot}}^*$9 and $\mu_{\mathrm T}(\zeta)$0, corresponding to an inertia-dominated regime spanning the classical random loose packing to random close packing range. For $\mu_{\mathrm T}(\zeta)$1, the system enters an adhesion-controlled universal regime in which $\mu_{\mathrm T}(\zeta)$2 becomes a single-valued function of $\mu_{\mathrm T}(\zeta)$3 and decreases systematically as $\mu_{\mathrm T}(\zeta)$4 increases. The loosest packing found numerically reached $\mu_{\mathrm T}(\zeta)$5 at $\mu_{\mathrm T}(\zeta)$6, while the coordination number $\mu_{\mathrm T}(\zeta)$7 collapsed onto a unique curve and dropped as adhesion strengthened (Liu et al., 2014).

A related simulation framework resolved this behavior into four regimes: an RCP regime for $\mu_{\mathrm T}(\zeta)$8, an RLP regime for $\mu_{\mathrm T}(\zeta)$9, an adhesion regime for $\Gamma$0, and an asymptotic regime for $\Gamma$1. In the adhesion regime the departure from random close packing was fitted as

$\Gamma$2

with $\Gamma$3 and $\Gamma$4. In the asymptotic regime the packing fraction converged toward $\Gamma$5, close to $\Gamma$6, and the coordination number approached $\Gamma$7 (Liu et al., 2015).

The mechanical interpretation is equally important. Adhesion modifies the critical sliding force,

$\Gamma$8

and enhances rolling resistance through adhesive contributions at the contact. As a result, particles can satisfy force and torque balance with fewer neighbors than in non-adhesive granular matter; many particles remain stable with only $\Gamma$9 or $\tau_C$0 neighbors in very loose chainlike structures (Liu et al., 2015).

This use of $\tau_C$1 is notable because it is not merely descriptive. It acts as the organizing variable for an adhesive branch in the $\tau_C$2–$\tau_C$3 jamming phase diagram. The associated statistical-mechanical theory yields an equation of state for adhesive loose packings and conjectures a maximally loose packing point at

$\tau_C$4

interpreted as the limiting end of the adhesive branch (Liu et al., 2014).

3. Energy-ratio adhesion numbers in capillary and bubble systems

In wall-attached bubble coalescence, the adhesion-related control parameter is not inertia-based but energy-based. The released surface energy is estimated as

$\tau_C$5

while the total adhesion energy is estimated as

$\tau_C$6

The normalized adhesion number $\tau_C$7 is then the ratio of these two scales (Demirkır et al., 9 Jan 2025).

The transition between bubble jumping and sticking is described by a global energy balance. At threshold, the translational kinetic energy after coalescence is approximately zero, so

$\tau_C$8

and, after normalization,

$\tau_C$9

For the previously unexplored low-effective-Ohnesorge regime, the critical adhesion threshold is $W_{\mathrm{ad}}$0, meaning that detachment typically occurs only when adhesion is below about $W_{\mathrm{ad}}$1 of the released surface energy. In the fitted combined dataset, $W_{\mathrm{ad}}$2 and $W_{\mathrm{ad}}$3, and the criterion correctly predicts about $W_{\mathrm{ad}}$4 of the jumping/sticking outcomes (Demirkır et al., 9 Jan 2025).

This formulation shows that an adhesion number may be meaningful only in combination with another normalized loss mechanism. Here that second parameter is the normalized viscous dissipation,

$W_{\mathrm{ad}}$5

with $W_{\mathrm{ad}}$6. The low-dissipation limit is adhesion-dominated; the small-adhesion limit is dissipation-dominated, with asymptotic threshold $W_{\mathrm{ad}}$7 (Demirkır et al., 9 Jan 2025).

Related capillary literature distinguishes carefully between adhesion work and retention force. For a droplet on a solid, the Young–Dupré work of adhesion is

$W_{\mathrm{ad}}$8

while the lateral retention force follows Furmidge’s law,

$W_{\mathrm{ad}}$9

and the normal capillary retention force is

$F/(\sigma d)$0

This separation is conceptually important: a capillary retention force per unit circumference is not automatically identical to the thermodynamic work of adhesion unless an additional assumption is imposed (Madrid et al., 2022).

4. Roughness, scale dependence, and the Tabor-number framework

For rough elastic solids, a single adhesion number is often insufficient because the relevant contact radius and effective adhesion energy vary with length scale. The scale-dependent Tabor number addresses this by using both the magnification-dependent curvature radius $F/(\sigma d)$1 and the roughness-renormalized interfacial energy $F/(\sigma d)$2. The resulting criterion is

$F/(\sigma d)$3

The central conclusion is that the same rough interface may be DMT-like at short length scales and JKR-like at large length scales (Persson et al., 2014).

This multiscale picture is tied to the roughness power spectrum. The effective radius is obtained from

$F/(\sigma d)$4

while the unresolved roughness amplitude enters through

$F/(\sigma d)$5

At higher magnification, shorter wavelengths are included, asperities become sharper, and $F/(\sigma d)$6 tends to decrease; this pushes the small-scale physics toward the DMT limit (Persson et al., 2014).

Hard-material adhesion experiments on rough diamond coatings reinforce the need for scale-aware descriptors. In sphere-on-flat pull-off tests with a $F/(\sigma d)$7-mm-diameter ruby sphere, roughness-dependent effective works of adhesion ranged from $F/(\sigma d)$8 to $F/(\sigma d)$9, whereas the geometry-independent intrinsic work of adhesion extracted from topography-resolved modeling was

$Ad=\frac{w}{2\rho_p U_0^2 r_p},$0

with adhesion range

$Ad=\frac{w}{2\rho_p U_0^2 r_p},$1

Filtering the roughness spectrum showed that the length scales with the strongest effect on pull-off force lay between $Ad=\frac{w}{2\rho_p U_0^2 r_p},$2 and $Ad=\frac{w}{2\rho_p U_0^2 r_p},$3 in lateral size (Thimons et al., 2021).

This suggests that, for rough hard contacts, any useful adhesion-number concept must be spectral rather than purely scalar. RMS roughness alone is insufficient; interaction range, yielding, and the specific roughness band sampled by the contact geometry all affect the macroscopic adhesion response (Thimons et al., 2021).

5. Systems in which no formal adhesion number is defined

A large fraction of adhesion research does not define a named adhesion number at all. Instead, it uses experimentally measurable quantities that serve as the physical basis from which such a number could be constructed.

In graphene mechanics, the primary quantity is the work of adhesion per unit area. A constant-$Ad=\frac{w}{2\rho_p U_0^2 r_p},$4 pressurized blister test measured adhesion energies of

$Ad=\frac{w}{2\rho_p U_0^2 r_p},$5

for monolayer graphene on $Ad=\frac{w}{2\rho_p U_0^2 r_p},$6 and

$Ad=\frac{w}{2\rho_p U_0^2 r_p},$7

for $Ad=\frac{w}{2\rho_p U_0^2 r_p},$8–$Ad=\frac{w}{2\rho_p U_0^2 r_p},$9-layer graphene. These values are reported as approximately four orders of magnitude larger than adhesion energies commonly found in micromechanical systems and comparable to solid/liquid adhesion energies. A separate morphology-based model for graphene on compliant patterned substrates uses the threshold

$Ad=\frac{\omega}{2\rho_p U_0^2 R},$0

to distinguish bonded, corrugated graphene from debonded, nearly flat graphene, thereby turning a morphology transition into an indirect adhesion measurement (Koenig et al., 2011, Zhang et al., 2011).

In graphene flake networks, the principal metric is the critical shear strength from button shear testing,

$Ad=\frac{\omega}{2\rho_p U_0^2 R},$1

where $Ad=\frac{\omega}{2\rho_p U_0^2 R},$2. Measured values include $Ad=\frac{\omega}{2\rho_p U_0^2 R},$3 MPa for Med Org, $Ad=\frac{\omega}{2\rho_p U_0^2 R},$4 MPa for Med OrgHMDS, $Ad=\frac{\omega}{2\rho_p U_0^2 R},$5 MPa for Low Org, and $Ad=\frac{\omega}{2\rho_p U_0^2 R},$6 MPa for High $Ad=\frac{\omega}{2\rho_p U_0^2 R},$7. The PMMA-only reference is much higher, $Ad=\frac{\omega}{2\rho_p U_0^2 R},$8 MPa, which supports the interpretation that BST is probing the graphene-related interface rather than PMMA alone. The study also defines a delamination-area measure,

$Ad=\frac{\omega}{2\rho_p U_0^2 R},$9

as a morphological complement to $w=2\gamma$0 (Estévez et al., 2024).

For soft interfaces, a FRET-based proxy of nanoscale contact plays the adhesion-metric role. The proposed C120/CDCF dye pair has $w=2\gamma$1 at $w=2\gamma$2 mM and a useful range of $w=2\gamma$3–$w=2\gamma$4 nm. FRET efficiency is

$w=2\gamma$5

and the reported experiments show that higher bonding pressure increases nanoscale contact, FRET signal, adhesion force, and separation energy. The authors explicitly state that adhesion force increases linearly with FRET efficiency (Simões et al., 2023).

For complementary rough PDMS elastomer surfaces, the wedge test yields a threshold work of adhesion

$w=2\gamma$6

The key result is non-monotonic dependence on roughness: the equilibrium crack length is minimum and the work of adhesion maximum at $w=2\gamma$7, while the equilibrium crack length is maximum and the work of adhesion minimum at $w=2\gamma$8 (Kumar et al., 13 Jul 2025).

Superhydrophobic droplet studies similarly avoid a universal adhesion number but use normalized forces. One experimentally validated relation is

$w=2\gamma$9

where $W_{a,\text{tot}}^*$00 is the pillar area fraction. In this framework, $W_{a,\text{tot}}^*$01 functions as an adhesion-number-like measure, and the average tensile force is reported as a better indicator of $W_{a,\text{tot}}^*$02 than the maximum force. A complementary numerical study treats the maximum force $W_{a,\text{tot}}^*$03 and detachment force $W_{a,\text{tot}}^*$04 as the main adhesion descriptors of the sawtooth force trace during recede (Kumar et al., 25 Nov 2025, Kumar et al., 25 Nov 2025).

6. Interpretation, limitations, and recurrent misconceptions

The main conceptual limitation is terminological: “adhesion number” is not a universal constant of materials. It is a context-dependent ratio or proxy whose denominator changes with the physics under study. In particulate deposition the competing scale is inertia; in bubble coalescence it is released surface energy and viscous dissipation; in rough-surface mechanics it is elastic compliance and interaction range through a scale-dependent Tabor construction (Liu et al., 2014, Demirkır et al., 9 Jan 2025, Persson et al., 2014).

A second recurrent misconception is to treat any adhesion-related measurement as equivalent to a work of adhesion. Several studies explicitly separate these notions. In capillary systems, Young–Dupré work, lateral retention force, normal retention force, and dynamic advancing or receding work per unit area are distinct quantities. In graphene flake networks, the measured shear strength reflects not only flake–substrate adhesion but also interflake adhesion and network morphology. In droplet-probe measurements on structured superhydrophobic surfaces, the maximum force is not necessarily the most representative adhesion metric because it is only one point in an intermittent stick-jump process (Madrid et al., 2022, Estévez et al., 2024, Kumar et al., 25 Nov 2025).

A third limitation is dimensionality. Many experimentally central adhesion descriptors are dimensional quantities: $W_{a,\text{tot}}^*$05 in $W_{a,\text{tot}}^*$06, $W_{a,\text{tot}}^*$07 in MPa, or $W_{a,\text{tot}}^*$08 in $W_{a,\text{tot}}^*$09 or $W_{a,\text{tot}}^*$10. These are not adhesion numbers unless normalized by an additional scale. The literature nevertheless shows that such dimensional measures often provide the physically decisive input for constructing a dimensionless criterion (Koenig et al., 2011, Simões et al., 2023, Kumar et al., 13 Jul 2025).

Taken together, the literature indicates that the most rigorous use of an adhesion number is as a reduced-order descriptor tailored to a specific competition: adhesion versus inertia, adhesion versus released capillary energy, or adhesion versus elastic roughness effects. Where no named number is introduced, the governing role is frequently played by an experimentally extracted work, strength, or normalized force that can serve as the numerator or core ingredient of such a construction (Liu et al., 2015, Zhang et al., 2011).