Peeling Surface Gravity: Mechanics & Criteria

• Peeling surface gravity is the process where gravitational forces overcome capillary or adhesive pinning, triggering the depinning of droplets or tapes.
• Quantitative analysis utilizes dimensionless groups like the Bond and Weber numbers to delineate transitions between gravity- and wind-dominated regimes.
• Experimental and theoretical models demonstrate how contact line geometry, viscoelastic dissipation, and interfacial fracture energies dictate the peeling behavior.

Peeling surface gravity refers to the specific role of gravitational forces in initiating or modulating the peeling or depinning of objects—typically droplets or adhesive films—from solid surfaces. The concept manifests in both the wetting dynamics of sessile liquid drops under combined capillarity, aerodynamic drag, and gravity, as well as in the mechanics of adhesive tape peeling driven solely by self-weight. Central to both lines of inquiry is the interplay between capillary or adhesive pinning and the force (or energy release rate) imparted by gravity, resulting in transitions between static adhesion and the onset of motion or depinning.

1. Dimensionless Groups and Peeling Criteria

In both liquid depinning and adhesive systems, surface gravity is captured quantitatively via the Bond number,

$Bo = \frac{\rho_d g D^2}{\sigma}$

for droplets ($\rho_d$: drop density; $D$: characteristic length; $g$: gravitational acceleration; $\sigma$: surface tension) and via the total load $F = Mg$ for adhesive rolls. The Bond number parameterizes the ratio of gravitational to capillary forces and controls the relative importance of gravity-driven peeling (gravity-forced drop depinning) versus other mechanisms.

For sessile drops (White et al., 2020), depinning is governed by a force balance per unit length along the contact line: $F_\gamma = F_\text{wind} + F_\text{grav}$ where $F_\gamma$ is the capillary pinning force, $F_\text{wind}$ the aerodynamic drag (parameterized by the Weber number $\mathrm{We}$), and $F_\text{grav}$ the downhill gravity component. Pinning and depinning thresholds are thus naturally expressed as critical values of $\mathrm{We}$ and $Bo$, often featuring empirical coefficients (e.g., $C_\gamma$, $C_w$) that encode the effects of contact line shape and drag geometry.

For adhesive tapes (Grzelka et al., 2021), peeling by gravity entails calculating the mechanical energy release rate $G$ per unit area, often via Rivlin’s relation: $G = \frac{F(1 - \cos{\theta})}{b}$ where $F$ is the external gravity load and $\theta$ the peel angle. This energy release competes with viscoelastic dissipation and interfacial fracture energies.

2. Pure Gravity-Forced Peeling and Depinning

In the absence of wind or auxiliary forcing, surface gravity alone can induce peeling or depinning. For drops on an inclined plate,

$$F_\text{grav} = C_g^* \rho_d g w_0^2 h_0 \sin\alpha$$

where $w_0$, $h_0$ are base width and height; $C_g^*$ is a geometric factor. The depinning transition occurs when gravitational torque overcomes capillary pinning: $$\rho_d V g \sin\alpha = \sigma w_0 \Delta\cos\theta,$$ which in nondimensional terms is $Bo\,AR\,\sin\alpha = \Delta\cos\theta$ (with $AR = w_0/h_0$).

For self-weight-driven peeling of adhesive rolls, the load is

$F = Mg, \quad m(s) = \rho b e (L_0 - s),$

leading to a nearly constant $G$ below significant roll mass loss: $G = \frac{Mg(1 - \cos{\theta})}{b}.$ Peeling proceeds quasi-statically if this $G$ exceeds bulk or interfacial resistance.

3. Mixed Forcing: Gravity, Wind, and Critical Regimes

In droplet systems, combined wind and gravity yield a linear force-balance equation in nondimensional form: $C_\gamma = C_w\,We + AR\,Bo\,\sin\alpha.$ Solving for the critical Weber number,

$$We_\text{crit}(Bo, AR, \alpha) = We_0 - m (AR\,Bo),$$

where $We_0 = C_\gamma / C_w$ and $m = (\sin\alpha)/C_w$. Importantly, $C_\gamma$ and $C_w$ change discontinuously between two forcing regimes:

Regime	$C_\gamma$	$C_w$	$We_0 = C_\gamma/C_w$	Slope $m = 1/C_w$
Gravity dominated	$\approx 1.323$	$\approx 0.042$	$\approx 31.5$	$\approx 24$
Wind dominated	$\approx 2.338$	$\approx 0.2955$	$\approx 7.9$	$\approx 3.38$

Empirically, piecewise linear fits—$We_\text{crit} = 7.9 - 3.38(AR\,Bo)$ for $We_\text{crit}>4$, $We_\text{crit}=7.9-24.0(AR\,Bo)$ for $We_\text{crit}<4$—collapse extensive experimental datasets. This dichotomy underscores a sharp transition as the dominant depinning mechanism shifts from gravity to wind (White et al., 2020).

For adhesives, the mechanical energy release rate $G$ can be altered by tuning the effective gravity (through planetary or centrifugal variations), but the principle persists: $G$ must exceed the sum of bulk viscoelastic dissipation and interfacial fracture energy for steady peeling (Grzelka et al., 2021).

4. Contact Line and Profile Evolution

Experimental studies reveal that gravitational and aerodynamic regimes impart distinct morphological signatures to sessile drops at the verge of depinning. White & Schmucker observed, via laser-speckle and side-view imaging:

• Wind-dominated regime: Drops exhibit a downstream-concave upwind face, with leading edge retreat to $0.87\,w_0$ and contact line curvature radius $\sim 0.39\,w_0$.
• Gravity-dominated regime: Drops elongate further ($0.98\,w_0$), and the advancing side radius is smaller ($\sim 0.31\,w_0$), with shapes closer to ideal spherical caps.

These findings suggest that the effective pinning coefficient $C_\gamma$ and drag coefficient $C_w$ are functions of evolving contact line geometry and curvature. However, the quantitative shift in $C_\gamma$ ($\sim 75\%$ across regimes) is not fully explained by observed shape changes, indicating that local microscopic receding-angle distributions play an essential, as yet unresolved, role.

5. Adhesive Systems: Energy and Rate Criteria under Gravity

The slow peeling of pressure-sensitive adhesives under gravity demonstrates that peeling velocities remain nonzero even at extremely low rates. In the viscoelastic regime, the empirical “Maugis–Barquins” law,

$G \propto V^n,\quad n \approx 0.166\text{–}0.507,$

captures the velocity dependence, with $n$ sensitive to humidity (and thus adhesive water content). All $G(V)$ data at constant temperature collapse onto a master curve through time–humidity superposition. The horizontal shift in rate axis, $a_{\rm RH}$, obeys a WLF-like expression dependent on adhesive water content: $$\log_{10}(a_{\rm RH}) = -D_1\,\frac{(W_c - W_{c,\rm ref})}{D_2 + (W_c - W_{c,\rm ref})},$$ with constants $D_1=0.87$, $D_2=1.39\%$.

At low humidity, a glass transition is approached, bulk viscous losses vanish, and fracture-like, rate-insensitive peeling emerges. The onset corresponds to reaching a critical $G_c\sim \Gamma$ (Griffith–Dugdale energy), with velocity governed by thermally activated bond-rupture kinetics.

6. Physical Mechanism of Gravity-Peeling

On the microscopic level, surface gravity in the absence of wind raises hydrostatic pressure at the receding edge of a droplet or adhesive, raising the local contact angle or promoting bond breaking at the interface. Once the gravitational torque or force exceeds the net pinning (or cohesive) resistance—set by $\Delta\cos\theta$ for drops, or fracture energy $G_c$ for adhesives—the system “peels” in a localized, often avalanche-like, event.

The key equilibrium in droplets is: $Bo\,AR\,\sin\alpha = \Delta\cos\theta,$ demarcating the precise conditions for gravity-induced depinning. In adhesives, the corresponding balance is $G \gtrsim G_c$.

7. Generalizations and Limitations

The analytic structure detailed above applies broadly to other adhesive and capillary systems under external load. Adjustments for peel angle variation, roll radius change, tape backing bending energy, and finite-strain rheology are possible extensions within adhesive mechanics. Material-specific master-curve shifts can be substituted for temperature, humidity, or solvent effects, enabling predictive modeling of peeling velocities under varying conditions (Grzelka et al., 2021).

Practical limitations of current models include approximations such as constant peel angle, negligible change in roll radius, and neglect of detailed microscopic contact-angle distributions. A more complete quantitative theory may require coupling mesoscale geometry evolution with bond-level mechanics and cohesive-zone modeling.

Taken together, peeling surface gravity provides a unified yet versatile basis for understanding the interplay between force, energy, and geometry in gravity-driven depinning of liquid drops and adhesive systems, spanning from high-Reynolds wind-forced regimes to quasi-static fracture-dominated peeling.