Evaporation and the coffee-ring effect for non-circular droplets
Madeleine Moore1 & Alex Wray2
1 University of Hull, 2University of Strathclyde
Contact: m.r.moore@hull.ac.uk, alexander.wray@strath.ac.uk
Introduction�Coffee rings
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A ‘coffee ring’ is the stain left behind after a droplet of coffee dries on a surface.
Introduction�Taking advantage of the coffee ring…
3
Aiding DNA mapping
Jing et al, PNAS 95:8046-8051 (1998)
Strengthening on the microscale
Chung et al, App. Phys. Lett. 105:261901 (2014)
Colloidal patterning
Choi et al, Langmuir 26(14):11690–11698 (2010)
Printing conductors
Layani et al, ACS Nano. 3(11):3547-3542 (2009)
Introduction�… or trying to fight it
4
Sam et al, Laser & Photonic Reviews 8(1):172-179 (2014)
In applications such as (AM)OLED screen manufacture, large arrays of non-circular droplets evaporate to form the RGB pixels.��
Understanding how the droplets evaporate, and the transport of any solute is a critical engineering challenge.
However, even the case of one irregularly-shaped droplet is non-trivial…
Introduction�Droplet geometry
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Saenz et al, Nature Comm. 8(1):1-9 (2017)
Droplet geometry is known to play a significant role in both evaporation and solute transport.
In the most common evaporation model – diffusive evaporation (which we will discuss shortly) – the evaporation rate is higher near parts of the contact line with high curvature.
Moreover, the coffee-ring effect is enhanced in these regions.
Introduction�Droplet geometry
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In fact, one can characterize* the shape of the growing coffee-ring explicitly
*Moore, Oliver & Vella, J. Fluid Mech., 940:A38 (2022)
solute mass profile
evaporative flux coefficient
local contact angle
gamma distribution PDF
droplet
solid
One of the major limiting factors in understanding the coffee ring effect is therefore determining how the droplet evaporates.
Problem configuration
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Consider a droplet of liquid evaporating from a solid substrate into the surrounding air.
Diffusive evaporation
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For the purposes of this analysis, we shall concentrate on a diffusive evaporation model.
In this process, evaporation is limited by the transport of vapour away from the liquid-air interface.
Convection effects are ignored and the evaporation is assumed to be slow, so that the whole process is quasi-steady.
The evaporative flux is then given by
Diffusive evaporation�Exact solutions
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This can be extended to hemispheres, hemi-ellipsoids and elliptical disks (see, for example, Kellogg (1929)), as well as spherical caps (Popov 2005).
This mixed boundary value problems occurs in a broad range of fields, including:
However, due to its complexity, very few analytical solutions exist.
Diffusive evaporation�Green’s function formulation
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Beyond these, it is extremely challenging to make any progress.
Analytically, things become intractable, while numerically, the singularity at the contact line is problematic.
A fruitful avenue of pursuit is to seek approximate or asymptotic solutions.
A sensible approximation is to assume that the droplet is thin, which is relevant in many applications – particularly evaporating droplets with a pinned contact line.
where
Diffusive evaporation�Green’s function formulation
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surface concentration (specified)
evaporative flux (desired)
We approach this by making use of Copson’s identity*
*Copson, Proc. Edin. Math. Soc. 8(1):14-19 (1947)
Diffusive evaporation�Green’s function formulation
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we may seek an asymptotic solution of the form
Diffusive evaporation�Leading-order solution
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Abel-type
Abel-type
L-type
We have two types of integrals that we can now invert.
L-type: Here we make use of the identity
Abel-type: Standard inversion formulae:
Diffusive evaporation�Leading-order solution
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We may apply these successively to invert the integral.
Iterating, we eventually return leading order solution,
which agrees with the classical Weber solution.
Diffusive evaporation�First order solution
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Proceeding to next order, it is helpful to write the leading-order solution in the asymptotically equivalent form
Then, writing
shifts singularity to correct location
ensures smoothness at origin
we find that
Diffusive evaporation�Solution
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Repeating this process, it follows* that the evaporative flux is given by
*about 20-odd pages of algebra later…
Diffusive evaporation�Solution
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Before validating, an important point to raise is that the leading-order solution
was used as an ansatz by Fabrikant (1986)* in his attempt at approximating solutions to this problem.
While this form has the correct singularity at the contact line and may be used to give good approximations of the total evaporation rate for a given flux, it is deficient in other aspects.
Notably, this form of the solution gives contours of the evaporative flux that are scaled versions of the contact line shape:
As we shall now see, this is decidedly far from reality.
*Fabrikant, ZAMP 36(4):616-623 (1987)
Validation
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We validate our asymptotic predictions by comparing to COMSOL simulations of the full mixed-boundary value problem.
Note the change in shape of the contours.
Validation
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Polygonal drops
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We motivated this analysis by considering arrays of square (or sometimes hexagonal) droplets seen in the manufacture of (AM)OLED screens.
As these are clearly far from circular, we might expect our analysis to not be applicable.
However, things go surprisingly well…
Consider a regular n-gon,
The goal is to find a Fourier expansion of this shape. However, the sharp corners cause problems.
until the maximal curvature does not exceed a certain tolerance. Typically we chose 10 dimensionless units.
Polygonal drops
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We then express the evaporative flux in the form
We may then truncate the series appropriately – this is shape dependent and depends on the decay of the coefficients – and compare to COMSOL simulations.
Polygonal drops
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Non-polygonal drops
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We can even extend this idea beyond regular polygons!
Wray & Moore (2023)
Solomon (1964)
Fabrikant (1986)
Aspect ratio
Application to the coffee-ring effect
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Mathematical model�Flow model
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When the droplet is thin, the liquid flow is governed by the lubrication equations
for
and the capillary and Bond numbers are defined by
at
Appropriate boundary conditions are boundedness at the origin alongside
and a specified initial droplet profile.
Mathematical model�Solute model
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We shall concentrate on the final deposit residue at the contact line, i.e. the total deposit swept to the contact line.
Since time is separable in this problem, the pathlines coincide with the streamlines.
Thus, it is straightforward to show that
Large droplets�Example: square droplet
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a)
b)
c)
d)
b) Pressure, streamlines and contours of the evaporative flux from the asymptotic model.
c) Residue density – the ‘coffee ring’ from the asymptotic model.
d) Residue density – the ‘coffee ring’ from the COMSOL results for a smoothed square.
Small droplets�Example: triangular droplet
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Saenz et al, Nature Comm. 8(1):1-9 (2017)
Flow streamlines calculated from the asymptotic model.
Free surface profile from the asymptotic model.
Summary & the future
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Summary
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We have derived a novel asymptotic solution for a classical problem in potential theory.
Our asymptotic model can be employed even in scenarios far from the theoretical limit, as we demonstrated by considering polygonal, rectangular and star-shaped droplets.
We demonstrated a particular application of the results for calculating the deposit residue after a solute-laden droplet evaporates.
Our model produces predictions in very good agreement with both COMSOL simulations and experimental data.
Future directions
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There are a number of different future directions, with several already under way.
�
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Thank you for listening!�
To find out more detail, see our (open access) JFM paper:
Or drop me a message: M.R.Moore@hull.ac.uk � https://twitter.com/Matheleine