Pendant Drop Method

• Pendant drop method is a tensiometric technique that infers interfacial tension from a gravity-deformed droplet’s shape using the Young–Laplace equation.
• It relies on a capillary–hydrostatic model with axisymmetric assumptions and extends to account for anisotropic tensions, evaporation effects, and volume constraints.
• Modern approaches combine high-resolution imaging, numerical fitting, and machine-learning inversion to enhance the precision and speed of surface tension measurements.

Searching arXiv for the cited pendant-drop papers to ground the article in current literature. arxiv_search(query="Pendant Drop Method Young-Laplace anisotropy evaporation machine learning pendant drop tensiometry", max_results=10, sort_by="relevance") The pendant drop method is a tensiometric technique in which the liquid–air or liquid–liquid interfacial tension is inferred from the static shape of a gravity-deformed droplet suspended from a capillary or needle. Its classical formulation treats the interface as axisymmetric about a vertical axis, with isotropic, spatially uniform, and constant surface tension $\gamma$, static equilibrium under gravity, negligible viscous and inertial effects, and a smooth interface with hydrostatic pressure in the bulk liquid and gas. In that setting, the droplet geometry is determined by the Young–Laplace relation and can be inverted from a single image. More recent work has shown that this classical picture remains powerful but is not universally sufficient: constant-volume constraints can produce position-dependent and anisotropic principal tensions, evaporation and condensation can bias measurements through thermal and compositional changes, and the inverse mapping from shape to material parameters can be accelerated by machine learning without replacing the underlying capillary–hydrostatic physics (Karr, 2019, Dekker et al., 10 Aug 2025, Kratz et al., 2020).

1. Definition and capillary–hydrostatic basis

A pendant droplet is a droplet hanging in air from a horizontal surface or capillary. The method exploits the fact that gravity deforms the droplet away from a sphere, while tangential surface tension at the capillary surface interface provides a balance of the internal and external pressures. The deformation encodes the ratio of hydrostatic forcing to capillarity, so the observed contour can be used to infer $\gamma$ when the density difference across the interface and the relevant geometric scale are known.

In the classical description, the local normal stress jump satisfies the Young–Laplace equation,

$$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$

with hydrostatic variation of the pressure jump along the interface. For a pendant droplet this is the central constitutive closure: geometry determines curvature, curvature determines pressure jump, and the comparison with measured shape determines $\gamma$. The method is therefore both a capillarity experiment and an inverse problem.

The literature distinguishes the free surface energy per unit area from the local surface tension. The free surface energy per unit area is commonly considered equal to the surface tension, but a variational treatment of axisymmetric droplets under constraints shows that this identification need not hold pointwise for the tangential tensions supported by the interface membrane (Karr, 2019).

2. Classical mathematical formulation

For an axisymmetric surface of revolution, it is standard to use arc length $s$ along the meridian, radius $r(s)$, vertical coordinate $z(s)$, and tangent angle $\psi(s)$ between the local tangent and the horizontal. The principal curvatures are

$$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$

and the mean curvature is

$H = \frac{\kappa_s+\kappa_\phi}{2}.$

With density difference $\gamma$0 between liquid and gas, the hydrostatic pressure difference is written

$\gamma$1

Under the classical assumption of constant isotropic $\gamma$2, the axisymmetric Young–Laplace relation becomes

$\gamma$3

The corresponding ODE system used in pendant-drop fitting is

$\gamma$4

This is the standard forward model integrated during shape fitting (Karr, 2019).

An alternative non-dimensionalization uses the apex curvature radius $\gamma$5 and the Bond number

$\gamma$6

In the Bashforth–Adams parametrization used for fitting, with arc length $\gamma$7, angle $\gamma$8, and coordinates $\gamma$9, the drop shape is written as

$$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$0

and the inferred surface tension follows from

$$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$1

This formulation is widely used in contemporary image-based pendant-drop tensiometry (Dekker et al., 10 Aug 2025).

3. Experimental realization and inverse estimation

In conventional practice, a pendant drop is formed at a capillary or needle, imaged, and converted into a digitized contour. The known or measured density difference $$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$2, gravity $$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$3, and a geometric scale such as the capillary diameter $$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$4 or apex radius $$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$5 are supplied to the inverse procedure. The contour is then compared with numerical solutions of the forward Young–Laplace equations until the least-squares mismatch is minimized, yielding the fitted material parameter.

A representative modern workflow combines optical calibration, contour extraction, and numerical fitting. One implementation uses a Nikon D850 with a 12× Zoom Navitar long-distance microscope, a glass cuvette with a bottom reservoir to passively set relative humidity, Gaussian blur with $$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$6 px, background and vignetting correction, subpixel edge detection, quartic correction for pincushion distortion, and camera tilt correction from a pendulum image. The fitting itself is two-step: a circle fit to the bottom $$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$7 of the interface yields $$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$8 and the center, and a Young–Laplace shape fit on each half of the contour yields $$\Delta p = \gamma \kappa = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right),$$9 and $\gamma$0, with final $\gamma$1. In that study, the threshold was chosen to minimize the RMS residual of the Young–Laplace fit, because the inferred $\gamma$2 could shift by $\gamma$3 if the threshold was not optimized (Dekker et al., 10 Aug 2025).

A more traditional numerical implementation uses fourth-order Runge–Kutta integration of the Young–Laplace ODEs from the apex, finite-difference Jacobians in parameter space, and steepest-descent updates of the dimensionless apex pressure and gravitational capillary parameter until residual parameter changes fall below a target step. In the reported implementation, convergence time per shape on an i7 CPU at $\gamma$4 was $\gamma$5–$\gamma$6 for the tested accuracy (Kratz et al., 2020).

Boundary conditions depend on the geometry of attachment. For a pendant drop attached to a capillary of diameter $\gamma$7, the apex conditions are

$\gamma$8

and the attachment condition is $\gamma$9. In a distinct experimental configuration, pendant drops of distilled water in air were formed from the inside diameter of a Teflon PFA capillary tube with inside diameter $s$0, giving the boundary condition $s$1 at the tube bottom (Kratz et al., 2020, Karr, 2019).

4. Variational reformulation and anisotropic surface tensions

A significant reinterpretation of the method arises from a calculus-of-variations treatment of the droplet as an incompressible, isothermal system with constrained volume. The total potential energy is written as the sum of surface energy and gravitational energy,

$s$2

subject to $s$3. In axisymmetric coordinates parameterized by $s$4, the functional is written

$s$5

where $s$6 is the free-surface energy per unit area and $s$7 is a Lagrange multiplier enforcing the constraint.

The Euler–Lagrange equilibrium relation is

$s$8

with

$s$9

If one further assumes $r(s)$0 and sets $r(s)$1, the classical Young–Laplace equation is recovered. The important distinction is that $r(s)$2 is maintained as a fluid property, whereas the tangential surface tensions in the interface membrane need not equal $r(s)$3 locally (Karr, 2019).

The generalized normal stress balance introduces principal tangential surface tensions $r(s)$4 and $r(s)$5,

$r(s)$6

and, for the reported static equilibria, explicit expressions follow:

$r(s)$7

These formulas show that when $r(s)$8, the principal tensions differ from $r(s)$9 and from each other, and vary with position through $z(s)$0 and $z(s)$1. If $z(s)$2, then $z(s)$3 everywhere, and the classical law is recovered.

The same work reports a quantitative example in which the smallest local tension occurs near the pendant top in the azimuthal direction, with $z(s)$4 for a larger-volume pendant. In that experiment, $z(s)$5 at room temperature, and the result established that local tensions can be lower than the free-surface energy under constant-volume constraints. The anisotropy disappears only at the pendant bottom, where $z(s)$6 and thus $z(s)$7 (Karr, 2019).

For such constrained states, the classical shape ODE is modified to

$z(s)$8

with $z(s)$9. This provides a generalized fitting framework when the constant-$\psi(s)$0 assumption is not justified.

5. Evaporation, humidity, and compositional drift

A second major qualification of the classical method concerns volatile or hygroscopic liquids measured in air. Pendant-drop tensiometry ordinarily assumes mechanical equilibrium, constant $\psi(s)$1 along the interface, and negligible internal flow. Evaporation or condensation during image acquisition violates these assumptions through evaporative cooling and composition changes, which alter both $\psi(s)$2 and the drop shape (Dekker et al., 10 Aug 2025).

For water at low relative humidity, the reported experiments measured a droplet temperature decrease of $\psi(s)$3 at steady state and an increase in inferred surface tension of $\psi(s)$4. Over that range, the empirical slope was $\psi(s)$5. At high RH near $\psi(s)$6, by contrast, the water drop temperature drifted by only $\psi(s)$7 and $\psi(s)$8 remained constant within uncertainty. The study concluded that an uncorrected $\psi(s)$9 bias exceeding $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$0 can dominate the intrinsic fit uncertainty (Dekker et al., 10 Aug 2025).

The same paper combined experiments with axisymmetric moving-mesh simulations. In the full simulations with evaporative cooling enabled, the apparent $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$1 from a Young–Laplace fit deviated by up to $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$2 from the initial $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$3 and by $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$4 from the surface-average $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$5, depending on RH. Thermal Marangoni circulation reduced interfacial temperature variation from $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$6 to $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$7 and improved the agreement between the dynamic shape and the static Young–Laplace approximation. This suggests that the principal measurement bias comes from changes in $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$8 rather than from gross shape distortion by the induced flow.

The same framework was extended to aqueous mixtures. Water/glycerol showed a monotonic decrease of $$\kappa_s = \frac{d\psi}{ds}, \qquad \kappa_\phi = \frac{\sin\psi}{r},$$9 with glycerol mass fraction, while water/1,2-hexanediol showed a strong surfactant-like drop in $H = \frac{\kappa_s+\kappa_\phi}{2}.$0 at low concentration, a minimum near $H = \frac{\kappa_s+\kappa_\phi}{2}.$1, and a slight increase of $H = \frac{\kappa_s+\kappa_\phi}{2}.$2 at higher concentration. Water/1,5-pentanediol exhibited an S-shaped $H = \frac{\kappa_s+\kappa_\phi}{2}.$3 with both a minimum near $H = \frac{\kappa_s+\kappa_\phi}{2}.$4 and a maximum near $H = \frac{\kappa_s+\kappa_\phi}{2}.$5, together with time-dependent $H = \frac{\kappa_s+\kappa_\phi}{2}.$6 in the cloudy composition range $H = \frac{\kappa_s+\kappa_\phi}{2}.$7–$H = \frac{\kappa_s+\kappa_\phi}{2}.$8. These results place the pendant drop method in direct contact with coupled thermal, solutal, and Marangoni phenomena.

A practical consequence is that RH and temperature control become part of the measurement model. A passive method uses a sealed glass cuvette with a bottom reservoir of the same liquid to set RH $H = \frac{\kappa_s+\kappa_\phi}{2}.$9, or anhydrous CaCl$\gamma$00 beads to obtain low RH, together with BME280 logging of chamber RH and temperature. For water-rich systems, the reported guidance is explicit: at RH $\gamma$01 with a passive reservoir and stable temperature, evaporation-induced bias is negligible; at RH below about $\gamma$02, several degrees of cooling can develop within seconds and a bias above $\gamma$03 should be expected unless corrected (Dekker et al., 10 Aug 2025).

6. Sensitivity, Worthington number, and machine-learning inversion

The inverse problem in pendant-drop tensiometry is not uniformly well conditioned. In the dimensionless formulation that uses capillary diameter $\gamma$04 as length scale, the independent parameters are the dimensionless apex pressure $\gamma$05 and the gravitational capillary parameter

$\gamma$06

The paper on machine learning emphasizes the role of the Worthington number,

$\gamma$07

as an indicator of shape sensitivity to $\gamma$08. High $\gamma$09, close to detachment, corresponds to shapes that are most sensitive to parameter changes; small $\gamma$10 corresponds to comparatively insensitive shapes and a more ill-conditioned inverse problem (Kratz et al., 2020).

The same work classifies axisymmetric shapes by

$\gamma$11

with $\gamma$12 for convex monotone widening shapes, $\gamma$13 for shapes with a single bulge, and $\gamma$14 for shapes with one bulge and one neck. Near the bifurcation boundaries in the shape diagrams, the volume–pressure and volume–$\gamma$15 curves have vertical tangents, indicating maximal sensitivity. In the experimentally common regime $\gamma$16, detachment and this bifurcation nearly coincide, so $\gamma$17 becomes a practical quality indicator for measurement design.

On that basis, a supervised deep-learning alternative was proposed. Instead of solving the nonlinear fit independently for each image, the method samples $\gamma$18 points along the non-dimensionalized contour, forms a fixed-length input vector, and maps it through a fully connected network with Leaky-ReLU activations and layer sizes

$\gamma$19

where the outputs are $\gamma$20. Synthetic training shapes are generated by forward integration of the Young–Laplace ODEs, initially uniformly in the $\gamma$21–$\gamma$22 region and later uniformly in the $\gamma$23–$\gamma$24 plane so that high-sensitivity shapes and $\gamma$25 states up to detachment are adequately represented (Kratz et al., 2020).

The reported quantitative comparison is specific. Conventional shape fitting required $\gamma$26–$\gamma$27 per shape on the tested CPU, whereas machine-learning inference required about $\gamma$28 per shape on a GTX 970, corresponding to an overall throughput increase of about $\gamma$29. On synthetic contours, the ML absolute errors were roughly one order of magnitude smaller than those of conventional shape fitting except in the very high-$\gamma$30 region, where sensitivity-aware training recovered the advantage. Training with Gaussian-blurred shapes and switching the loss from MSE to MAE improved robustness to contour noise. These results do not change the governing physics; they change the computational cost and, under the tested conditions, the precision of the inversion.

7. Validity domain, diagnostics, and relation to other methods

The pendant drop method is valid when its structural assumptions are valid. Across the cited works, those assumptions are consistent: axisymmetry, static equilibrium, smooth interfaces, incompressible bulk fluids, negligible inertial and viscous effects, and a homogeneous, constant surface tension when the classical model is used. The method becomes unreliable when non-axisymmetry, evolving adsorption, Marangoni stresses, evaporation-driven thermal drift, or constraint-induced anisotropy materially affect the shape (Karr, 2019, Dekker et al., 10 Aug 2025, Kratz et al., 2020).

Several common misconceptions follow from treating the classical model as universal. One is that the free-surface energy per unit area and the local surface tension are always identical; the constrained variational analysis shows that local principal tensions can differ from the free-surface energy and from each other. Another is that evaporation during a short acquisition is negligible; for water at low RH, measurable cooling and a surface-tension shift can occur within $\gamma$31. A third is that a successful constant-$\gamma$32 fit necessarily validates the classical assumptions; the anisotropy study notes that identical pendant geometries can occur with different $\gamma$33, so a constant-$\gamma$34 interpretation can be suspect if geometry is unchanged while the hydrostatic reference pressure differs (Karr, 2019, Dekker et al., 10 Aug 2025).

The practical diagnostics are therefore model-based. Classical Young–Laplace analysis is trustworthy when droplets are formed under pressure control with slow equilibration and small volume changes so that $\gamma$35, when tension gradients are small so that $\gamma$36, and when ambient conditions suppress evaporation and condensation. In contrast, volume-constrained formation histories, variable upper meniscus heights, or large deviations between inferred principal tensions signal that the generalized energy-based framework is more appropriate. The same logic motivates complementary checks with methods such as the Wilhelmy plate or spinning drop when an independent estimate of $\gamma$37 is required (Karr, 2019).

Within those limits, the pendant drop method remains a central technique for interfacial tensiometry because it links capillary geometry, hydrostatics, and imaging in a quantitatively explicit way. Its modern form is not a single algorithm but a family of inference procedures built around the same geometric balance: classical Young–Laplace fitting when constant isotropic $\gamma$38 is a valid approximation, constrained variational analysis when local principal tensions are history-dependent and anisotropic, and humidity-aware or temperature-corrected protocols when evaporation or condensation couples the shape to $\gamma$39.