Contents

Polynomial fitting method

Pronunciation: /ˌpɒlɪˈnəʊmiəl ˈfɪtɪŋ ˈmɛθəd/

The polynomial fitting method finds the contact angle by fitting a low order polynomial to the drop edge close to the contact point and taking the slope of that curve where it meets the baseline. It uses only the local shape, so it works when the whole drop is not axisymmetric.

Also called

Polynomial fit Tangent fitting method Local profile fitting

Related terms

Drop shape analysis (DSA) Drop volume

Topics

Contact angle Image analysis Method
Book a 15 minute demo Read the contact angle measurement guide

At a glance

What polynomial fitting does with a drop image, and when to choose it over a full profile fit.

What it is

Software detects the drop edge and the baseline. It then fits a short curve, often a second order polynomial, to the edge points just above each contact point. The contact angle is the angle of that curve's tangent at the baseline.

  • Left and right angles are found separately
  • No assumption about the overall drop shape
  • Fast, so it suits video and dynamic angles

Worked example

Near the left contact point, with the baseline at z = 0, the fit gives z = 0.70 x − 0.40 x² (in mm).

  • Slope at the contact point: dz/dx = 0.70 − 0.80 x = 0.70 at x = 0
  • θ = arctan 0.70 = 35.0°
  • A moderately hydrophilic surface

Students practise this kind of fit in labs from the surface science teaching curriculum hub.

How it is measured

Polynomial fitting is an analysis step applied to a sessile drop image or video. Image capture, baseline placement and reporting are covered in the contact angle measurement guide.

Expert view

“What a liquid responds to is only the outermost few ångström of a surface: the skin, not the bulk. Change that skin and you change the wetting, even if the material underneath is identical.”

Dr. Alidad Amirfazli, Co-founder and Scientific Advisor, Droplet Lab

What the terms mean

Where a local polynomial fit works well, and the symbols used.

Symmetric drop, no needle

Both a polynomial fit and a full Young-Laplace fit work. The Young-Laplace fit uses the whole outline and is less affected by local defects.

Needle in the drop or asymmetric drop

The overall shape no longer matches an axisymmetric model. A local polynomial fit at each contact point still gives a reliable angle.

Very low or very high angles

Few edge points lie close to the contact point, or the edge turns past vertical. Fit choices and baseline placement then matter most.

Only the edge points close to the contact point are used. The tangent of the fitted curve at the baseline gives the contact angle.

θ Contact angle at the contact point, through the liquid, degrees
z(x) Fitted edge height above the baseline as a function of horizontal position, mm
a0, a1, a2 Polynomial coefficients returned by the fit
xc Horizontal position of the contact point, mm

Key equations

The fit and the step from slope to angle.

Equation: z(x) = a0 + a1 x + a2 x²

Variable legend:

  • z: height of the edge above the baseline (mm)
  • x: horizontal position (mm)
  • a0, a1, a2: fitted coefficients

Interpretation: A short curve that follows the drop edge near the contact point closely enough to give its slope.

When to use: Use it on edge points close to each contact point, separately for the left and right sides.

Assumptions:

  • The edge is smooth over the fitted region
  • The baseline is correctly placed

Common mistakes:

  • Fitting too many points far from the contact point
  • Using a high order polynomial that follows pixel noise

Equation: θ = arctan |dz/dx| at x = xc

Variable legend:

  • dz/dx: slope of the fitted curve at the contact point
  • θ: contact angle (°), for angles below 90°

Interpretation: The steeper the edge rises from the baseline, the larger the contact angle.

When to use: Use it for angles below 90°. Above 90° the edge leans back over the contact point, so the angle is 180° minus the acute tangent angle.

Assumptions:

  • Angle measured through the liquid

Common mistakes:

  • Reporting the acute tangent angle for a drop above 90°
  • Fitting z as a function of x where the edge is near vertical; many programs fit x as a function of z there

How to read it

What a polynomial fit result tells you, and what to check before you trust it.

Polynomial fitting is precise when the image is sharp and the baseline is right. Checking the instrument first, as described in the contact angle goniometer calibration guide, separates fit issues from setup issues. Real world spread across many users is visible in the dataset of client collected wetting measurements.

From polynomial fit result to meaning

If you see It usually means Check next
Left and right angles differ by several degrees Tilted stage, uneven surface or baseline offset Stage level, baseline line on the image
Angle jumps between video frames Edge noise or too few fitted points Image focus, lighting, fit region
Polynomial and Young-Laplace results differ on a clean, symmetric drop A local defect at one contact point, or baseline error Repeat on a fresh drop and position
Unstable values near 180° or below 10° Few edge points close to the contact point Higher magnification, careful baseline placement

How it is measured

The fit in four short steps. The contact angle measurement guide has the full procedure.

1

Capture the image

Record a sharp, backlit side image or video of the drop.

2

Set the baseline

Detect or place the baseline where the drop meets the solid.

3

Fit near each contact point

Fit a low order polynomial to the edge points just above each contact point.

4

Read the angles

Take the tangent of each fitted curve at the baseline and report left, right and mean angles.

Common misconceptions

Where polynomial fitting results are most often misread.

Polynomial fitting is less accurate than Young-Laplace fitting

On synthetic drops, both fits can be very accurate. Young-Laplace uses the whole outline and is more robust to local defects, while polynomial fitting handles needles and asymmetric drops. Choose by drop geometry, not by habit.

A higher order polynomial is always better

Higher orders follow pixel noise and can bend near the end of the fitted region, exactly where the angle is read. Low orders with a sensible number of points near the contact point are the usual choice in published methods.

The fit fixes a poor baseline

The angle is read where the curve meets the baseline. A baseline a few pixels too high or low shifts that point and changes the angle, most of all for high angle drops.

Related terms

Where polynomial fitting sits among drop shape methods.

Broader term

  • Drop shape analysis (DSA)
  • Contact angle

Related terms

  • Drop volume
  • Sessile drop method
  • Advancing and receding contact angle
  • Goniometer

Often confused with

  • Young-Laplace equation (full profile fit)
  • Spherical cap (circle) fitting

FAQ

Short answers to the questions people ask most about the polynomial fitting method.

When should I use polynomial fitting instead of a Young-Laplace fit?

Use polynomial fitting when the drop is not axisymmetric: a needle in the drop during advancing and receding measurements, a drop on a tilted plate, or a drop pinned unevenly. For a symmetric drop at rest, a Young-Laplace fit uses the whole outline and is less sensitive to a local defect.

What order of polynomial is used for contact angle fitting?

Low orders are usual. A second order polynomial is common, and published work has examined how the fit settings affect accuracy at different angles. The aim is a curve that follows the real edge near the contact point without chasing pixel noise.

Which instruments use polynomial fitting?

Many drop shape analysis programs offer it alongside circle and Young-Laplace fits. The Dropometer smartphone drop shape analysis kit applies drop shape analysis with automatic edge and baseline detection. Its drop shape method is validated in peer review against a KRUSS DSA100E, with a published accuracy of 0.35°.

References

Primary sources and further reading.

1. Atefi, E., Mann, J. A., and Tavana, H. (2013). A robust polynomial fitting approach for contact angle measurements. Langmuir, 29(19), 5677 to 5688. https://doi.org/10.1021/la4002972
2. Chen, H., Muros-Cobos, J. L., and Amirfazli, A. (2018). Contact angle measurement with a smartphone. Review of Scientific Instruments, 89(3), 035117. https://doi.org/10.1063/1.5022370
3. Huhtamäki, T., Tian, X., Korhonen, J. T., and Ras, R. H. A. (2018). Surface-wetting characterization using contact-angle measurements. Nature Protocols, 13(7), 1521 to 1538. https://doi.org/10.1038/s41596-018-0003-z
4. Contact angle measurement guide
5. Drop shape analysis (DSA)
6. Drop volume