Role of the free surface in particle deposition during evaporation of
colloidal sessile drops
Hassan Masoud and James D. Felske
Department of Mechanical and Aerospace Engineering,
State University of New York at Buffalo,
Buffalo, New York 14260, USA
Deposition patterns of particles suspended in evaporating colloidal drops are determined
by the flow fields within the drops. Using analytically determined velocities, particle
motions are then tracked in a Lagrangian sense. It is found that the majority of particles
intersect the free surface as it recedes. Such “capture” of particles by the free surface is
found to be the major mechanism in establishing the deposition pattern. Patterns are
calculated for wetting and non-wetting drops whose contact lines are either pinned or
freely moving during evaporation. The distribution of evaporative flux which drives the
flows is taken to be that engendered by gas-phase diffusion. The theoretical results are
found to agree favorably with available experimental data.
I.
INTRODUCTION
Recently, the problem of sessile drop evaporation has found prominence in relation to
the deposition of particles that occurs during the drying of colloidal drops. The topic is of
high current interest because of the use of evaporating drops in depositing molecules (e.g.
DNA or proteins) onto substrates in high through-put genomic or proteomic assays,
colloidal particles into ordered structures for potential use directly, or for the templating
of ordered structures. In addition, a particular deposition pattern – the ring – and
phenomena related to it have important implications for: drug discovery [1], and the
manufacture of novel electronic and optical materials [2,3], including thin films and
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coatings [4-7]. Apart from ring patterns, applications include: buckling instability and
skin formation by deposition from polymer solutions [8-10], and evaporation of liquid
drops on cool or hot surfaces [11].
The deposition pattern produced depends upon the flow within the drop. In the
absence of a surface tension gradient at the free surface, the flow pattern is driven by the
combination of the evaporative flux distribution, the shape of the free surface, and the
behavior of the contact line. For instance, flow can be towards, away or both towards and
away from the contact line under different combinations of these factors. Thermal
boundary conditions and surfactant concentration at the free surface are equally important
when Marangoni flow is present.
Several studies, including our previous analytical analyses [12,13], have been
conducted focusing on the flow inside an evaporating sessile droplet without Marangoni
flow. In addition, Deegan et al. [14], Popov [15], Fischer [16], Hu and Larson [17] and
Widjaja and Harris [18] have investigated the particle deposition during sessile drop
evaporation. Deegan [14] and Popov [15] neglected the vertical velocity component and
instead considered the behavior of the vertically averaged radial velocity at small contact
angles. In their analyses 100 % of the solute particles were swept radially to the contact
line of the drop. Fischer [16] obtained the fluid velocity in the limit of the lubrication
theory. Neglecting particle diffusion, he calculated the particle concentration distribution
due solely to convective mass transfer. Hu and Larson [17] simulated Brownian
dynamics to predict deposition. They determined the deposition distribution by freezing
the location of any particle that came into contact with the substrate. Widjaja and Harris
[18] calculated particle deposition profiles from an Eulerian point of view. They found
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that the deposition pattern is influenced by both the convective and diffusive mass
transfer of particles in the bulk liquid as well as by the deposition rate along the substrate.
In the present study, deposition patterns resulting from the flow within evaporating
colloidal drops are investigated from the Lagrangian point of view when convection is
the primary mode of particle transfer. It is shown that the free surface of the drop plays a
major role in defining the distribution pattern. Patterns are calculated for wetting and
non-wetting drops whose contact lines are either pinned or freely moving during
evaporation.
II.
MODELING AND ASSUMPTIONS
The droplets considered in the present study are initially ~ 1mm from the axis of
symmetry to the contact line ( R ). For water drops of this size evaporating under room
conditions, the characteristic velocity in the drop is ~ 1  m / s with a corresponding
Reynolds number of ~ 103 [19]. For millimeter size water drops the Bond and capillary
numbers are, respectively, Bo  ~ 0.04 and Ca  ~ 10 8 . Hence, surface tension is the
dominant influence on the droplet‟s shape, and the droplet becomes a spherical cap.
The droplets are assumed to contain dilute suspensions of non-volatile molecular or
colloidal solutes which are initially uniformly dispersed in the fluid. The solutes are
assumed to move at the same velocity as the solvent but, unlike the solvent, they do not
evaporate.
Particle motion is tracked in a Lagrangian sense. When they are deposited, it is
assumed that the deposits do not interfere with the geometry of the drop or the flow field
within it. Therefore, in the absence of a surface tension gradient at the free surface, the
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analytical solutions previously derived [12,13] for the velocity distributions are
employed.
Particle diffusion is neglected with respect to convective transport since, as pointed
out by Hu and Larson [17], it takes much longer for a micron size particle to diffuse
across the height of a mm-size drop than it takes for the radial flow to carry the particle to
the free surface.
Also, the particles are considered to be simple spheres which do not interact with one
another.
Particles may experience the following phenomena: „capture‟ by the substrate,
„capture‟ by the free surface, and transport by the bulk fluid. Regarding „capture‟ by the
substrate, it is noted that the fluid velocity approaches zero as the substrate is neared.
Therefore, without enhancement of diffusion or attractive forces, particles in solution will
not be captured by the substrate. On the other hand, it will be seen that „capture‟ by the
free surface occurs for the majority of particles. Once captured, a particle moves with the
velocity of the surface wherein the tangential component carries it towards the contact
line. It is assumed that in nearing the contact line, when a particle reaches
(rparticle / R  1)2  ( z particle / R  1) 2  0.01 it remains fixed at that position. Those particles
in the bulk fluid which approach the contact line are treated in the same manner. The
deposition thickness is computed at the end of evaporation.
Two different models of contact line motion are considered. The first, based on the
experimental observation of Hu and Larson [17], takes the droplet contact line to be
pinned until the contact angle is reduced to a critical value after which the contact line
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freely recedes. The second model allows the contact line to always freely move (never
pinned).
FIG. 1: Initial distribution of particles in the drop ( Acs* / Ntotal  4 104 ). Particles entering the
indicated zone near the contact line stay there.
In comparing particle tracking results for drops of different sizes (contact angles) the
area initially allotted to a particle should be the same. Computationally this requires a
common value of the ratio of the dimensionless cross sectional area ( Acs*  Acs / R 2 ) to the
total number of particles ( N total ). In Fig. 1 the uniform initial spacing corresponding to
Acs* / N total  4 104 is shown for contact angle of 60 . This value was used in all of the
present results. That this value was small enough to yield behavior independent of its
value was verified in a number of cases by computing for Acs* / N total  104 .
For a pinned contact line, the rate of change of the contact angle is known as a
function of time. In these cases, the simulations were performed by incrementing the
contact angle rather than the time. For an unpinned d contact line, the rate of change of
the radius is known as a function of time. In this case, simulations incremented this radius
rather than time.
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III.
RESULTS AND DISCUSSION
Figure 2 shows the deposition patterns corresponding to the first model of contact line
behavior. This model pins the contact line until the contact angle is reduced to a critical
value. When this value is reached, the drop becomes unpinned and the contact radius
freely recedes (at fixed contact angle) for the remainder of evaporation. All patterns
predicted by this model are ring-like. It is seen that as the initial contact angle increases,
the fraction of particles deposited at the contact line (and consequently the thickness of
the deposit) increases. This behavior follows from first noting that surface and bulk flows
are both toward the contact line when it is pinned. Therefore, since drops having larger
initial contact angles provide more time for particle motion, a larger fraction of the
particles will finally deposit at the contact line.
FIG. 2: Fraction of particles deposited corresponding to different initial contact angles ( 30 , 60 , 90
and 120 ). First model: contact line pinned until contact angle reaches 3 and subsequently recedes at this
angle.
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FIG. 3: For (a)–(c), the initial contact angle is 40 ; (a) and (b) show particle depositions
corresponding, respectively to viscous and inviscid flow; (c) compares the fraction of particles captured by
the free surface as a function of time (equivalently, c (t ) ) in viscous and inviscid flows. (d) Fraction of
particles captured by the free surface in viscous flow for different initial contact angles ( 30 , 60 , 90 and
120 ). In all figures the droplet contact line is pinned until the contact angle is reduced to 3 whereupon
the contact radius recedes at constant contact angle.
Figure 3 compares deposition behavior computed for flows considered to be either
viscous or inviscid. From previous analyses [12,13] it was demonstrated that inviscid and
viscous flows are similar near the free surface but are somewhat different near substrate
(due to the no slip condition in viscous flow). Because of the differences in the near
substrate flow behavior, it might be surmised that viscous and inviscid flows will produce
distinctly different deposition patterns. This, however, is not the case, as it can be seen
from Fig.3 (a) and (b). The deposition patterns are actually quite similar and both agree
well with the experimental data of Hu and Larson [17] (Fig. 4). The reason why inviscid
and viscous depositions agree so well is that it is not the near substrate flow behavior
which controls deposition but rather the flow behavior near the free surface. This stems
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from the behavior illustrated in Fig. 3(d) which shows that the majority of particles are
„captured‟ by the free surface during the time the contact line is pinned. Of those particles
captured by the free surface, most are carried down to the contact line by the tangential
fluid velocity on the free surface. Figure 3(c) also illustrates this behavior. Therefore,
since the number of particles captured by the free surface is about the same for viscous
and inviscid flows, and since near the free surface the flow fields are similar in both
cases, then deposition patterns will be similar whether the flow is treated as viscous or
inviscid.
FIG. 4: Fraction of particles deposited as a function of radial position for two initial contact angles
( 30 and 60 ). Second model: contact line unpinned during evaporation.

Figure 4 illustrates particle deposition using the second model of contact line behavior –
free to move (unpinned) during evaporation. In this case the fractional deposition is
slightly decreasing from the axis out to the initial droplet radius. Rings are not produced
when contact lines are not pinned.
IV.
CONCLUSION
The present results provide additional understanding of the nature of particle
deposition during the drying of colloidal drops. It was shown that the droplet free surface
plays a major role in defining the particle distribution pattern. The mechanism of particle
capturing by the free surface was found to dominate the transport of particles to the
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substrate. This behavior and the resulting deposition patterns were explored using
analytical solutions for the velocity field along with a simple, yet accurate, tracking of
particle positions. The shape of the deposition pattern was found to depend on the model
used for pinning of the contact line. Finally, deposition patterns were found to be similar
for viscous and inviscid models of the flow. This shows that near-substrate transport
plays only a minor role in particle deposition. It is therefore possible to employ the more
readily calculated inviscid solution in order to obtain reliable, semi-quantitative results.
ACKNOWLEDGMENT
The importance of this problem was brought to our attention by Prof. R. C.
Wetherhold.
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