Capillarity-Weighted Volume Fraction

• Capillarity-weighted volume fraction is defined as an effective descriptor that adjusts classical volume measures based on capillary-screening and grain-size distribution.
• It systematically influences material behavior across granular media, soft aerated solids, and capillary channels by altering the mechanical and transport response.
• This weighting mechanism incorporates factors like bubble stiffness, free-energy minimization, and geometric occupancy to improve predictions in heterogeneous systems.

Capillarity-weighted volume fraction denotes a class of quantities in which a geometric phase fraction, volume fraction, or effective occupied volume is modulated by capillarity-dependent physics rather than counted uniformly. Its most explicit recent definition is a screened, distribution-aware replacement for the classical solid volume fraction in capillarity-screened Darcy flow, where the relevant fraction depends on grain size and capillary length (Tantardini et al., 19 Sep 2025). In adjacent literatures, closely related constructions appear in different forms: as gas fraction interpreted through a capillary number in soft aerated solids, as an equilibrium partial saturation selected by free-energy minimization in cornered channels, as interface-associated subvolumes in straight capillaries, and as capillarity-weighted energetic or measure-theoretic objects in variational droplet and soap-film problems (Ducloué et al., 2013, Yu et al., 2020, Yelkhovsky et al., 2018, Maggi et al., 2015, King et al., 2019, Dipierro et al., 2020). The term therefore does not name a single universal scalar across the literature; it identifies a recurring principle that capillarity changes the mechanical, transport, or variational relevance of a given amount of material.

1. Terminological range and scope

Several papers use the underlying idea of a capillarity-conditioned fraction without introducing the exact same variable. In stationary random granular media, the capillarity-weighted volume fraction is defined explicitly as a screening-dependent average over grain sizes. In soft aerated materials, by contrast, no single variable with that name is introduced; the relevant control is the pair of gas volume fraction and capillary number. In wetting equilibrium and pore-scale capillary flow, the nearest analogues are a free-energy-selected saturation and geometry-based occupancy variables. In variational capillarity, the corresponding objects are effective adhesion parameters or weighted measures rather than classical fractions (Tantardini et al., 19 Sep 2025, Ducloué et al., 2013, Yu et al., 2020, Yelkhovsky et al., 2018, Maggi et al., 2015, King et al., 2019, Dipierro et al., 2020).

Setting	Primary quantity	Role of capillarity
Stationary random granular media	$\phi_\lambda$	Screening weights size-resolved volume fraction
Soft aerated materials	$(\phi, Ca)$	Bubble stiffness weights mechanical effect of gas fraction
Rectangular channel wetting	$s^*$	Free-energy minimization selects partial saturation
Straight capillary two-phase flow	$V$, $a_{\rm film}$, $a_{\rm bulk}$	Curvature and pressure determine occupied subvolume
Small-volume droplet in container	$\tau_0$	Adhesion weights asymptotic droplet geometry
Soap-film capillarity limit	$$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$	Collapse induces multiplicity weighting
Fractional capillarity	${\mathcal C}_{s,\sigma}$ and extension energy	Nonlocal adhesion weights boundary interaction

The unifying feature is that raw Euclidean volume is not, by itself, the operative descriptor. Instead, capillarity alters which parts of a microstructure, interface, or phase distribution are mechanically active, spectrally visible, or energetically preferred.

2. Screened, distribution-aware volume fraction in granular media

The most literal definition appears in the framework for capillarity-screened Darcy flow in stationary random, polydisperse granular media (Tantardini et al., 19 Sep 2025). There the classical solid volume fraction is

$\phi=\langle \chi(\mathbf x)\rangle,$

with $(\phi, Ca)$0 the phase indicator. The capillarity-weighted volume fraction is introduced as

$(\phi, Ca)$1

where $(\phi, Ca)$2 is grain size, $(\phi, Ca)$3 is the volume-fraction density by size, and

$(\phi, Ca)$4

This definition is distribution-aware because the weighting depends on the screening length $(\phi, Ca)$5 and on the grain-size distribution.

The construction is motivated by the screened Green’s function of a modified-Helmholtz or Yukawa operator. In Fourier space the screening weight is

$(\phi, Ca)$6

Using the covariance $(\phi, Ca)$7 of the phase indicator and its spectral density $(\phi, Ca)$8, the screened correlation content is

$(\phi, Ca)$9

Approximating the covariance by size-diagonal pieces $s^*$0, the size weight becomes

$s^*$1

In dilute spherical form, the paper gives the proxy

$s^*$2

with

$s^*$3

This makes explicit that screened capillarity acts as a spatial low-pass filter: sizes contribute according to how much of their spectral content survives below the cutoff $s^*$4.

The paper states that $s^*$5 are the distribution-aware replacements for $s^*$6 when screening strongly attenuates sub-$s^*$7 features. The companion descriptor is the screened integral range

$s^*$8

or equivalently

$s^*$9

These quantities enter the variance model

$V$0

the refined apparent screening

$V$1

and the supercell-size rules

$V$2

and, for cubic cells,

$V$3

In the classical limit,

$V$4

Thus, in this setting, capillarity-weighted volume fraction is a formal screened descriptor with direct consequences for homogenized screening strength, apparent capillary decay length, and representative-volume selection.

3. Gas fraction weighted by bubble capillarity in soft aerated materials

In soft aerated solids, the phrase does not denote a standalone variable. The elastic response is governed by two dimensionless parameters: the gas volume fraction $V$5 and a capillary number

$V$6

where $V$7 is the shear modulus of the bubble-free matrix, $V$8 is the surface tension, and $V$9 is the bubble radius (Ducloué et al., 2013). The dimensionless modulus is reported as

$a_{\rm film}$0

Within this formulation, a capillarity-weighted volume fraction means that the mechanical effect of $a_{\rm film}$1 is conditioned by bubble stiffness through $a_{\rm film}$2, rather than by $a_{\rm film}$3 alone.

The dilute limits show the polarity of this weighting. In the hole limit,

$a_{\rm film}$4

so bubbles soften the material. In the rigid-inclusion limit,

$a_{\rm film}$5

so bubbles stiffen it. The same gas volume fraction can therefore either reduce or increase the modulus depending on capillarity. The experiments identify a crossover near

$a_{\rm film}$6

for which

$a_{\rm film}$7

At this value, adding bubbles is nearly non-perturbative.

The paper rationalizes the crossover by modeling each bubble as an equivalent elastic sphere with effective modulus

$a_{\rm film}$8

Since

$a_{\rm film}$9

the bubble has the same effective elasticity as the matrix at the crossover. Capillarity therefore renormalizes the gas inclusion into an effective elastic object.

Using a Mori-Tanaka or homogenization approach, the effective shear modulus is given by

$a_{\rm bulk}$0

This expression reduces to the dilute limits, predicts the crossover at $a_{\rm bulk}$1, and describes the full dependence on both $a_{\rm bulk}$2 and $a_{\rm bulk}$3. The experimental observations reported in the paper are consistent with this interpretation: changing bubble radius at fixed matrix changes $a_{\rm bulk}$4, larger bubbles soften the material more, changing matrix modulus at fixed bubble size also changes $a_{\rm bulk}$5, systems with similar $a_{\rm bulk}$6 but different surface tension behave similarly, and the data agree well with $a_{\rm bulk}$7. In particular, behavior approaches the hole limit for $a_{\rm bulk}$8, while the response is nearly unchanged for $a_{\rm bulk}$9–$\tau_0$0.

The central implication is narrow but important: in this literature, capillarity does not redefine volume fraction algebraically; it changes the constitutive meaning of a given gas fraction by altering bubble deformability.

4. Equilibrium saturation and geometric occupancy in capillary channels

A different capillarity-conditioned fraction appears in wetting equilibrium in a nearly rectangular channel, where the relevant quantity is the fluid fraction of the partially filled, finger-like region, denoted $\tau_0$1 (Yu et al., 2020). The channel contains a fully filled bulk region with $\tau_0$2 and a partial-filling region with $\tau_0$3. The paper emphasizes that $\tau_0$4 is selected by free-energy minimization and is directly tied to capillary pressure. With free-energy density per unit axial length $\tau_0$5 and total cross-sectional area $\tau_0$6, the capillary pressure is

$\tau_0$7

The coexistence condition for the equilibrium partial saturation is

$\tau_0$8

This is the tangency condition between the free-energy curve and the secant joining $\tau_0$9 to the fully filled state.

The analysis distinguishes three geometries for the partially filled region: four separate corner fingers, two merged fingers, and a full circular meniscus. In the four-finger regime, the free energy has the form $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$0 and yields an explicit $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$1. For a square cross-section, the paper gives $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$2. In the two-finger regime, $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$3 can become much larger. The paper reports that for $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$4, changing $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$5 from $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$6 to $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$7 increases $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$8 from nearly zero to about $$\mu=H^n\!\llcorner(\partial^*E)+2H^n\!\llcorner(K\setminus\partial^*E)$$9. This is the basis for the conclusion that a small deviation from a perfect rectangle can drastically alter the equilibrium partial-filling fraction. The model is compared with two experiments: for the Keita et al. case, ${\mathcal C}_{s,\sigma}$0 and ${\mathcal C}_{s,\sigma}$1 give ${\mathcal C}_{s,\sigma}$2, compared with an observed interface saturation of about ${\mathcal C}_{s,\sigma}$3; for the Seck et al. case, ${\mathcal C}_{s,\sigma}$4 and nearly perfect rectangular geometry yield ${\mathcal C}_{s,\sigma}$5, consistent with very thin corner columns.

In straight-capillary two-phase flow, the relevant occupancy variables are more local and geometric than thermodynamic (Yelkhovsky et al., 2018). The paper does not define a capillarity-weighted volume fraction, phase fraction, or saturation in that terminology. Instead, the interface evolution is governed by the volume-balance equation

${\mathcal C}_{s,\sigma}$6

where ${\mathcal C}_{s,\sigma}$7 is a subvolume of either phase. The cross-sectional liquid occupancy in rounded corners is

${\mathcal C}_{s,\sigma}$8

and the bulk area is

${\mathcal C}_{s,\sigma}$9

Capillarity enters through the pressure jump

$\phi=\langle \chi(\mathbf x)\rangle,$0

through curvature-dependent film area, through capillary-pressure gradients that can induce counterflow, and through the dynamic contact-angle law

$\phi=\langle \chi(\mathbf x)\rangle,$1

The phase distribution is represented by a discrete set of interface variables: meniscus position, anchor curvature radii, and film tip position. In this context, the closest analogue to a capillarity-weighted fraction is the curvature-controlled partition of the cross-section into bulk and corner-film occupancies.

These two channel-scale literatures share a specific feature: the relevant “fraction” is not imposed as a constitutive constant but selected or evolved through capillary geometry, pressure, and interfacial energy.

5. Variational and geometric measure-theoretic formulations

In variational capillarity, the phrase acquires a more abstract meaning. For droplets in a bounded container, the minimized quantity is the Gauss free energy

$\phi=\langle \chi(\mathbf x)\rangle,$2

for finite-perimeter sets $\phi=\langle \chi(\mathbf x)\rangle,$3 of prescribed volume $\phi=\langle \chi(\mathbf x)\rangle,$4 (Maggi et al., 2015). The paper does not introduce a weighted volume fraction as a new scalar, but it identifies the capillarity-weighted effective datum

$\phi=\langle \chi(\mathbf x)\rangle,$5

In the small-volume regime,

$\phi=\langle \chi(\mathbf x)\rangle,$6

and minimizers concentrate near boundary points where $\phi=\langle \chi(\mathbf x)\rangle,$7 is minimal and converge, after flattening and rescaling, to the sessile cap $\phi=\langle \chi(\mathbf x)\rangle,$8. Here the “weighting” operates through adhesion and perimeter, not through a phase-fraction field.

For soap films modeled as liquid regions of small prescribed volume, the capillarity problem introduces a weighted measure that explicitly distinguishes genuine liquid boundary from collapsed structure (King et al., 2019). The relaxed global functional is

$\phi=\langle \chi(\mathbf x)\rangle,$9

and the associated measure is

$(\phi, Ca)$00

The density is $(\phi, Ca)$01 on $(\phi, Ca)$02 and $(\phi, Ca)$03 on $(\phi, Ca)$04. As the prescribed volume $(\phi, Ca)$05,

$(\phi, Ca)$06

and generalized minimizers converge to a Plateau minimizer counted with multiplicity two. In this setting, the capillarity-weighted object is not a fraction of material by volume; it is a measure-theoretic record of whether the film remains a genuine liquid boundary or collapses onto a multiplicity-two skeleton.

A nonlocal analogue appears in fractional capillarity (Dipierro et al., 2020). The fractional capillarity energy is

$(\phi, Ca)$07

and the associated extension functional is

$(\phi, Ca)$08

The scale-invariant quantity

$(\phi, Ca)$09

satisfies a boundary monotonicity formula, and blow-ups of local minimizers are minimizing cones. In the planar case there is only one possible fractional minimizing cone, determined by the fractional Young’s law. The relevant weighting here is again not a scalar volume fraction but a nonlocal boundary interaction that measures how phase volume near the boundary contributes to capillarity.

These variational formulations broaden the meaning of capillarity-weighting from effective fractions in transport or mechanics to capillarity-dependent selection principles in geometric analysis.

6. Conceptual synthesis, limiting regimes, and recurrent misconceptions

Taken together, these works suggest that capillarity-weighted volume fraction is best understood as a family of capillarity-conditioned descriptors rather than a universally fixed variable (Tantardini et al., 19 Sep 2025, Ducloué et al., 2013, Yu et al., 2020, Yelkhovsky et al., 2018, Maggi et al., 2015, King et al., 2019, Dipierro et al., 2020). In the most literal case, the descriptor is a screening-weighted phase fraction $(\phi, Ca)$10. In soft composites, the analogous weighting is carried by the capillary number $(\phi, Ca)$11, so the same gas fraction can correspond to hole-like softening, rigid-inclusion strengthening, or nearly neutral response. In corner wetting, the relevant fraction is the coexistence saturation $(\phi, Ca)$12 selected by the free-energy landscape. In pore-scale capillary dynamics, the effective occupancy is encoded by subvolumes and curvature-dependent cross-sectional areas. In geometric variational settings, capillarity weights volume indirectly through adhesion parameters, nonlocal interactions, or multiplicity in limiting measures.

Several recurring misconceptions are resolved by this comparison. First, the phrase is not uniformly defined across capillarity research: multiple papers explicitly do not introduce a variable literally called capillarity-weighted volume fraction. Second, the classical unweighted fraction is often recovered in a limiting regime rather than rejected outright; for screened granular media, $(\phi, Ca)$13 as $(\phi, Ca)$14. Third, equal raw fractions need not imply equal response. The granular-media formulation states that two microstructures with the same $(\phi, Ca)$15 but different size distributions can produce different capillary responses, and the soft-solid formulation shows that the same $(\phi, Ca)$16 can either stiffen or soften a material depending on $(\phi, Ca)$17. Fourth, geometry can dominate the effective fraction: in the nearly rectangular channel, very small changes in cross-sectional shape can move the system between qualitatively different coexistence states with sharply different $(\phi, Ca)$18.

A plausible synthesis is that capillarity-weighting always identifies which part of a nominal phase amount is actually relevant to the governing physics. The weighting variable may be a capillary length, a spectral filter, a bubble stiffness scale $(\phi, Ca)$19, a free-energy tangency condition, an adhesion minimum, or a geometric multiplicity. What remains invariant across these formulations is the rejection of a purely geometric reading of volume fraction in favor of one conditioned by capillary sensitivity, interfacial curvature, or variational structure.