Bendo-Capillary Length in Elastic Interfaces

• Bendo-Capillary Length is defined as the characteristic scale where capillary forces balance elastic bending, marking the crossover from weak deformation to strong wrapping.
• In microfiber–droplet experiments, its expression L₍BC₎ = √(Er³/γ) organizes bending behavior and predicts winding thresholds based on material properties and droplet size.
• In flexible-walled channels, bendocapillary scaling determines instability wavelengths via parameters such as γ, B, and channel geometry, offering a predictive framework for capillary-induced deformations.

Bendo-capillary length denotes a characteristic scale that emerges when capillary forcing is balanced by elastic bending. The term is not fully standardized across the cited literature. In the microfiber–droplet problem, the explicit quantity is the bending elastocapillary length $L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$, which organizes bending and winding of a flexible fiber by a droplet (Schulman et al., 2016). In a flexible-walled channel, the corresponding scale appears as a critical wavelength or characteristic in-plane length derived from the balance between capillary pressure and wall bending stiffness, notably through $k_c$, $\lambda_c$, and $L_y$ (Bradley et al., 2022). By contrast, some multiscale wetting papers explicitly state that they do not define a standalone quantity named “Bendo-Capillary Length,” even though they analyze coupled capillary and small-scale interfacial physics (Fricke et al., 2023). This suggests that the expression is best understood as a generic label for capillarity–bending crossover scales rather than as a single universally fixed symbol.

1. Terminology and principal formulations

The literature represented here uses the term in two closely related but not identical ways. One usage is localized and object-based: a flexible slender body is bent by a liquid interface, and a single length scale marks the crossover between weak deformation and strong capillary wrapping. The other usage is distributed and instability-based: a liquid deforms a flexible confinement, and the relevant scale is the wavelength at which capillary feedback and bending resistance balance.

Context	Characteristic scale	Stated role
Microfiber around a droplet	$L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$	Crossover for substantial bending and spontaneous winding
Flexible-walled channel	$$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$, $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$, $$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$	Instability wavelength and characteristic in-plane bendocapillary scale
Capillary rise with slip	no specific quantity named “Bendo-Capillary Length”	Multiscale coupling without a dedicated bendocapillary variable

These formulations are all explicit in the cited papers (Schulman et al., 2016, Bradley et al., 2022, Fricke et al., 2023). A plausible implication is that “bendo-capillary length” is not a single canonical symbol, but a category of lengths attached to a capillarity–bending competition.

2. Mechanical balance underlying bendocapillary scaling

In the microfiber problem, the competition is between the fiber bending modulus

$B=\frac{\pi E r^4}{4}$

and capillary forcing from a droplet of surface tension $\gamma$. The paper states that capillary forces act on the fiber over a droplet-contact length scale, and that the competition between capillary work and bending energy produces

$k_c$0

The physical interpretation given is direct: for $k_c$1, deformation is weak; for $k_c$2, the fiber bends strongly; and for sufficiently large droplets the fiber can wind around the droplet (Schulman et al., 2016).

In the flexible-channel instability, the relevant balance is not a static wrapping criterion but a capillary–elastic feedback in which wall deformation modifies local channel width, width modifies meniscus curvature, and curvature modifies liquid pressure. The paper identifies the key ingredients as capillary pressure set by $k_c$3 and $k_c$4, elastic bending set by $k_c$5, channel geometry through $k_c$6, $k_c$7, and meniscus position $k_c$8, and liquid volume through $k_c$9 or $\lambda_c$0. The resulting critical wavenumber is

$\lambda_c$1

with associated characteristic wavelength

$\lambda_c$2

The same paper introduces the dimensionless bendability parameter

$\lambda_c$3

interpreted as the ability of the typical capillary pressure to bend the walls, and the reduced aspect ratio

$\lambda_c$4

In this setting, larger $\lambda_c$5 promotes instability, larger $\lambda_c$6 stabilizes, larger $\lambda_c$7 stabilizes, and larger $\lambda_c$8 strongly increases instability tendency because the wall deflection scales with a longer lever arm and scales as $\lambda_c$9 (Bradley et al., 2022).

3. Flexible fibers and the bending elastocapillary length

The most explicit named bendocapillary scale in the cited literature is the bending elastocapillary length of a microfiber in contact with a droplet,

$L_y$0

The experiments involve polystyrene (PS) fibers with $L_y$1 GPa and radii $L_y$2, and styrene-isoprene-styrene (SIS) fibers with $L_y$3 MPa and radii $L_y$4. The droplet is glycerol, and the experiments are performed far below the capillary length, so gravity is neglected (Schulman et al., 2016).

A minimal Euler–Bernoulli beam model is used in the contact region. With wetted length $L_y$5, meniscus size $L_y$6, and distributed load

$L_y$7

the beam equation is

$L_y$8

Using the stated boundary conditions, the deflection profile inside the wetted region is

$L_y$9

From this, the outer slope gives the bending-angle relation

$L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$0

while the local central curvature gives

$L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$1

The data collapse when plotted against $L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$2, and the paper states that bending depends primarily on $L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$3, not on $L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$4 or $L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$5 separately (Schulman et al., 2016).

The same length also organizes the transition from bending to winding. A beam-theory estimate gives the winding threshold at

$L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$6

for the PS system. An energetic criterion writes the free-energy change per unit length as

$L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$7

with winding when $L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$8, or equivalently

$L_{\mathrm{BC}}=\sqrt{E r^3/\gamma}$9

where

$$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$0

Using measured wetting angles, the predicted thresholds are $$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$1 and $$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$2. The measured PS threshold is $$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$3, while SIS gives $$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$4; the latter differs from the simple energetic prediction because the experimental geometry includes extra bending outside the contact region (Schulman et al., 2016).

4. Bendocapillary wavelength in flexible-walled channels

In a flexible-walled channel partially filled by a droplet, the bendocapillary scale is a wavelength selected by instability rather than a droplet size selected by static deformation. The paper states that a perturbation in the meniscus position changes wall deformation, which changes channel width, which in turn changes meniscus curvature and liquid pressure. This feedback can grow when capillary-induced pressure overcomes the stabilizing effect of in-plane curvature of the interface (Bradley et al., 2022).

The characteristic scale is derived through the critical wavenumber

$$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$5

or, equivalently, the wavelength

$$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$6

The paper also defines

$$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$7

described as the characteristic in-plane length scale for the instability. In the paper’s summary, this is the dimensional version of the bendocapillary wavelength scale (Bradley et al., 2022).

Several consequences follow directly from the stated scaling. Because $$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$8, increasing liquid volume moves the system toward shorter unstable wavelengths. The growth rate is highly volume-sensitive, with the dimensional estimate

$$k_c \sim \left(\frac{\gamma \cos^2 \theta\, x_m^3}{B H^3}\right)^{1/2}$$9

which the paper emphasizes as a very strong dependence on liquid volume through $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$0 or $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$1. The same paper also states that wetting and non-wetting configurations are susceptible in the same channel, because the leading-order critical scale depends on $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$2, so the scaling is the same for both signs of wettability (Bradley et al., 2022).

In this formulation, the bendocapillary length is inseparable from geometry and loading history. It is not a material constant alone, because it depends on $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$3, $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$4, $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$5, $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$6, and especially the equilibrium meniscus position $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$7.

5. Distinction from the classical capillary length

The cited papers sharply distinguish bendocapillary scales from the classical capillary length. In meniscus problems around cylinders and particles, the capillary length is

$$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$8

and its function is to separate gravity-free and gravity-limited regimes, not bending-dominated and bending-resisted regimes (Tang et al., 2018, Tang et al., 2018).

For a vertical circular cylinder in a coaxial cylindrical container, the capillary length is the crossover scale between a container-controlled regime and a gravity-controlled regime. When $$\lambda_c \sim 2\pi \left(\frac{B H^3}{\gamma \cos^2 \theta\, x_m^3}\right)^{1/2}$$9, gravity is negligible and the meniscus height scales approximately as $$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$0; more explicitly,

$$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$1

When $$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$2, the height becomes independent of $$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$3 and follows the Derjaguin–James law,

$$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$4

with $$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$5. The crossover region is identified as roughly

$$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$6

and the two asymptotic predictions intersect at $$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$7 (Tang et al., 2018).

A closely parallel crossover appears for a small particle at a liquid-vapor interface. There, for $$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$8, the meniscus rise or depression grows as $$L_y=\left(\frac{B H^3}{\gamma \cos^2 \theta L^3}\right)^{1/2}$$9, while for $B=\frac{\pi E r^4}{4}$0 it saturates to the Derjaguin–James form. The effective spring constant likewise changes from

$B=\frac{\pi E r^4}{4}$1

for $B=\frac{\pi E r^4}{4}$2 to the saturated form

$B=\frac{\pi E r^4}{4}$3

for $B=\frac{\pi E r^4}{4}$4 (Tang et al., 2018).

The same gravity–surface-tension length also organizes capillary attraction between partially submerged vertical cylinders. There the capillary length is

$B=\frac{\pi E r^4}{4}$5

reported as $B=\frac{\pi E r^4}{4}$6 for the mineral oil used. The crossover between pressure-dominated and surface-tension-dominated attraction occurs at a separation of around half of a capillary length, and the model is noted to be unreliable for very large separations beyond about $B=\frac{\pi E r^4}{4}$7 (Rieser et al., 2014).

These papers therefore treat the capillary length as a gravity-screening or gravity-crossover length. That role is conceptually different from the bendocapillary role, where the competition is capillary forcing versus elastic bending.

6. Related scales, scope, and terminological cautions

Not every capillarity-controlled characteristic length is a bendocapillary length. In capillary rise dynamics with Navier slip, the paper contrasts macroscopic capillary-rise scales with microscopic wetting scales and states explicitly that it does not introduce a specific quantity named “Bendo-Capillary Length.” Instead, the dynamics are organized by $B=\frac{\pi E r^4}{4}$8, $B=\frac{\pi E r^4}{4}$9, the slip length $\gamma$0, and the parameter

$\gamma$1

In that setting, the dominant dependence on slip is logarithmic in $\gamma$2, for example through

$\gamma$3

and the fitted effective friction parameter depends approximately linearly on $\gamma$4 (Fricke et al., 2023).

Other problems introduce still different capillary-controlled lengths. In steady capillary jets, the natural breakup length is nondimensionalized with

$\gamma$5

a pressure–capillarity length used in the scaling law

$\gamma$6

with $\gamma$7 and $\gamma$8 (Ganan-Calvo et al., 2019). In colloidal capillary interactions, the capillary length

$\gamma$9

sets the range over which interface deformations decay and therefore determines whether the attraction is effectively logarithmic or exponentially screened (Dominguez et al., 2010).

A plausible implication is that the most precise use of “bendo-capillary length” is restricted to scales explicitly generated by a capillarity–bending balance, such as $k_c$00, $k_c$01, or $k_c$02. The cited literature does not support using the term as a synonym for the classical capillary length, the slip length, or a generic capillary scaling length.