A Linear-Regression-Based Method for Determining Surface Tension from Variation in Interfacial Curvature

We develop a new linear-regression-based method for determining surface tension, γ, from interfacial curvature. Across a static fluid–fluid interface, γ is balanced by the difference in hydrostatic pressure, ΔP. The balance is described by the Young–Laplace relation: ΔP = γ (k1 + k2), where k1 + k2 is the sum of the principal interfacial curvatures. Along the interface, ΔP varies linearly with elevation. It has been assumed that, even when there is surfactant at the interface, γ is constant along the interface. If this assumption is correct, then, according to the Young–Laplace relation, k1 + k2 must also vary linearly with elevation. For images of a constrained sessile droplet and a captive bubble, each with an interfacial dipalmitoyl phosphatidylcholine (DPPC) monolayer that is compressed to varying degrees, we determine the offset in ΔP from that at a reference elevation, ΔP – ΔPo, and k1 + k2 at interfacial points of different elevations. We find that k1 + k2 indeed varies linearly with elevation. Thus, if we plot ΔP – ΔPo versus k1 + k2, we obtain a linear plot with a slope equal to γ. We develop and make available an algorithm for determining γ by linear regression. And we assess accuracy by comparison to γ values determined by axisymmetric drop shape analysis. For a droplet or bubble with a DPPC monolayer, over a γ range of 2–70 mN/m, the absolute value of the difference in γ determined by the alternative methods averages 0.30 ± 0.37 mN/m. Further, while we apply the method to axisymmetric interfaces and make use of axisymmetry to determine the curvature from two-dimensional images, the underlying theory does not require axisymmetry. With 3D volumetric imaging, the method could potentially be extended to more complex interfacial geometries.