Quasi-Stationary Evaporation of a Small Droplet on a Flat Substrate: New Analytical Solution

What is it about?

The study of a sessile liquid droplet evaporating on a flat surface is of great importance for physicochemical, technical, and medical applications. New analytical expressions have been proposed for the vapor density, evaporation flux density, and total evaporation flux per unit time for a slowly evaporating small axially symmetric droplet placed onto a flat substrate at an arbitrary value of the contact angle ranging from 0 to 180°. When deriving the expressions, the solution of the Laplace equation well-known in electrostatics for a flat wedge has been used. The solution has been transformed by the method of inversion on a sphere into a solution for a lens in bipolar coordinates. The new expressions are mathematically equivalent to previouslyproposed equations in toroidal coordinates [Popov, Yu.O., Phys. Rev. E, 2005, vol. 71, p. 036313]; however, in the bipolar coordinates, the evaporation flux density has a simpler form of a single integral of a combination of elementary functions, thus being advantageous from the computational point of view. A new expression has also been proposed for the evaporation flux density in polar coordinates and graphic constructions have been performed for the dependences of the evaporation flux density on the polar angle at different values of the droplet contact angle.

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Photo by David von Diemar on Unsplash

Photo by David von Diemar on Unsplash

Why is it important?

A liquid droplet evaporating on a flat substrate is an important object for both theoretical simulation (evaporation dynamics, hydrodynamics, self-organization of a solute, etc.) [1–4] and numerous applications (printing technologies, functionalized coatings, medical diagnostics, etc.) [5–12]. Three main problems may be distinguished when describing an evaporating solution droplet: [13, 14]: (1) evaporation of a solvent from the droplet surface into ambient air (external problem), (2) hydrodynamic flows in the droplet bulk (internal problem), and (3) dynamics of particles (molecular or colloidal) in the droplet taking into account interparticle, particle–surface, and particle– solvent interactions. The first problem (evaporation) is of primary importance, since it is the primary driving force for the self-organization of the matter in the droplet.

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