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Abstract

Soft microparticles are deformable ("squishy") microscale entities that are ubiquitous in nature and technology. Unlike macroscopic particles, the force response of microparticles under external loading depends on both bulk material properties and interfacial tension, allowing force measurements to probe these properties. Traditional devices such as the biomembrane force probe, optical and magnetic tweezers, the atomic force microscope and the microcantilever tensiometer each have limitations in force range, accessible particle size, reliability of attachment, or commercial availability. Consequently, even simple microparticle subclasses such as Newtonian and viscoelastic microdroplets remain challenging to characterize accurately at the single-particle level. The cantilevered capillary force apparatus (CCFA) overcomes these limitations by fixing soft microparticles via capillary rim pinning or gentle suction, imposing controlled quasi-static or oscillatory deformations and resolving the resulting forces with high accuracy. However, analytical frameworks for interpreting CCFA force signals and inferring material properties under these deformations have been lacking. To address this, we first perform a small-strain regular perturbation expansion of the Young–Laplace equation, the force response, and the bridge volume-conservation constraint to obtain the leading nonlinear force and shape of an initially spherical Newtonian microdroplet quasi-statically deformed while pinned between two coaxial substrates. Comparison with numerical solutions demonstrates excellent agreement and substantially extends the range of validity of earlier analytical models, enabling accurate interfacial tension measurements over a wider range of deformations. Second, we develop an asymptotic model for the small-strain oscillatory deformation of an initially cylindrical Maxwell viscoelastic microdroplet pinned between two coaxial substrates. Regular perturbation in strain amplitude and aspect ratio gives a force response with a static capillary part and a dynamic part comprising in-phase contributions from bulk elasticity and interfacial curvature, and an out-of-phase contribution from bulk viscosity. In time-resolved CCFA measurements, the force offset yields the interfacial tension, while the amplitude and phase lag yield the storage and loss moduli. Together, these models provide a unified analytical framework for extracting interfacial and bulk material properties of single microdroplets using the CCFA, and they can be extended to more complex soft microparticles and related instruments.