### TL;DR
Robert Deegan is a professor of physics at the Univer...
Here's a video by Robert Reegan on the effect
Capillary flow as the cause of ring stains from dried liquid drops
Robert D. Deegan, Olgica Bakajin, Todd F. Dupont, Greg Huber, Sidney R. Nagel, Thomas A. Witten
James Franck Institute, 5640 South Ellis Ave, Chicago, IL 60637, USA
typeset July 15, 1997
When a spilled drop of coffee dries on a solid
surface, it leaves a dense, ring-like stain along
the perimeter (Figure 1a). The coffee—initially
dispersed over the entire drop—becomes concen-
trated into a tiny fraction of it. Such ring deposits
are commonplace wherever drops containing dis-
persed solids evaporate on a surface. Thus ring
deposits influence printing, washing and coating
processes [1–5]. They provide a potential means
to write or deposit a fine pattern onto a surface.
Here we ascribe the deposition to a previously un-
explored form of capillary flow: the contact line
of the drying drop is pinned so that liquid evapo-
rating from the edge must be replenished by liq-
uid from the interior (Figure 2). The resulting
outward flow can carry virtually all the dispersed
material to the edge. This mechanism predicts
a distinctive power-law growth of the ring mass
with time—a law independent of the particular
substrate, carrier fluid or deposited solids. We
have verified this law by microscopic observations
of colloidal fluids.
FIG. 1. a) A 2-cm-diameter drop of coffee containing
one-weight-percent solids has dried to form a perimeter
ring, accentuated in regions of high curvature. b) Video
micrographs showing dispersion of fluorescent one-micron
polystyrene spheres in water during evaporation, as de-
scribed in the text. Multiple exposure shows different
times in different colors to indicate the motion. Earliest
time (blue) is 3 sec before latest time (red).
Our qualitative observations show that rings form for a
wide variety of substrates, dispersed materials (solutes),
and carrier liquids (solvents), as long as (1) the solvent
meets the surface at a nonzero contact angle, (2) the con-
tact line is pinned to its initial position, and (3) the sol-
vent evaporates. In addition, we found that mechanisms
typically responsible for solute transport—surface ten-
sion gradients, solute diffusion, electrostatic, and gravity
effects—are negligible in ring formation. Based on these
findings, we have identified the minimal ingredients for
a quantitative theory. The phenomenon is due to a ge-
ometrical constraint: the free surface, constrained by a
pinned contact line, squeezes the fluid outward to com-
pensate for evaporative losses. We first describe the main
features and results of our theory and then show our ex-
perimental tests of its validity.
a
J
r
h
v
_
c
b
R
FIG. 2. Mechanism of outward flow during evapora-
tion. Pictures a and b show an increment of evaporation
viewed in cross section. Picture a shows the result of
evaporation without flow: the droplet shrinks. Picture
b shows the compensating flow needed to keep the con-
tact line fixed. Picture c defines quantities responsible
for flow. Vapor leaves at a rate per unit area J(r). The
removed liquid contracts the height h(r) vertically, va-
cating the vertically striped region in a short time ∆t.
The volume of this striped region is equal to the volume
removed by J. But in the shaded annular region the red-
striped volume is smaller than the volume removed by J
there (red arrows). Thus liquid flows outward to supply
the deficit volume: fluid at r sweeps out the blue-striped
region in time ∆t. Its volume is the deficit volume; its
depth-averged speed is ¯v(r).
Figure 2 sketches the factors leading to outward flow
in a small, thin, dilute, circular drop of fixed radius R
slowly drying on a solid surface. The evaporative flux
J(r) reduces the height h(r) at every point r. If there
were no flow, the evaporation would alter the height pro-
1
file as sketched in Figure 2a. At the perimeter, all the
liquid would be removed and the drop would shrink. But
the radius of the drop cannot shrink, since its contact line
is pinned. To prevent the shrinkage, liquid must flow out-
ward as in Figure 2b. The complete flow profile ¯v(r) can
be found as sketched in Figure 2c. The height profile
must maintain the spherical cap shape dictated by sur-
face tension. Thus during a short time ∆t the vertically
striped region must be removed from each point r of the
surface. This is different from the amount removed from
that point owing to evaporation, shown in color in Figure
2a. Radial flow must make up for this difference. If the
evaporative flux J(r) is known, the flow velocity ¯v(r)is
thereby determined.
The evaporative flux has a universal form that depends
only on the shape of the drop ( for example see: [6,?]).
In the evaporation process liquid molecules interchange
rapidly between the surface and the adjacent air, so that
this air is saturated with vapor. Since the air at infinity is
not saturated, the vapor diffuses outward. At the surface
of the drop (which is the only region where we need to
calculate the evaporation current) the vapor quickly es-
tablishes a steady-state concentration profile φ(~r) which
obeys the steady-state diffusion equation ∇
2
φ =0. At
infinity, φ is the ambient concentration φ
∞
. At the drop
surface φ is fixed at the saturation concentration φ
s
. The
derivitive at the surface gives the desired evaporating flux
J(r)=−D∇φ, where D is the diffusivity of the vapor in
air.
We may find the flux J by solving an equivalent electro-
static problem [8], wherein φ is an electrostatic potential
and the drop, with its fixed potential, is a conductor. (On
the substrate surface beyond the drop there is no flux, so
that the normal derivitive of φ is zero there. That sur-
face is in effect a reflection plane of symmetry and the
opening angle of the conducting wedge is twice the con-
tact angle that the drop makes with the substrate.) The
sharp wedge-shaped boundary of the drop and its reflec-
tion leads to a diverging normal derivitive (electric field
or evaporative flux) as r approaches the contact line. For
a drop with contact angle θ
c
, this divergence has the form
[9]: J(r) ∼ (R − r)
−λ
, where λ =
π−2θ
c
2π−2θ
c
. As the contact
angle decreases towards zero, λ increases towards 1/2.
The initial deposition over times much shorter than the
drying time depends sensitively on this diverging flux.
To replace the diverging flux requires a diverging veloc-
ity near the perimeter: ¯v ∼ J ∼ (R − r)
−λ
.Wenow
consider the mass of solute, M(r, t) beyond distance r
from the center at time t. Near the contact line the ini-
tial mass M(r, 0) is proportional to the volume of this
wedge shaped region: M(r, 0) ∼ (R − r)
2
. All this mass
will be entrained in the ring in the time t required for a
point starting at r
t
to move to the contact line:
t =
Z
R
r
t
dr/¯v ∼ (R − r
t
)
1+λ
, (1)
since ¯v(r) ∼ (R − r)
−λ
. Substituting for r
t
in terms of t
into M(r
t
, 0) yields a power law for the early time growth
of the ring: M (R, t)=M(r
t
,0) ∼ t
2/(1+λ)
.
At late times, the theory predicts complete transfer of
the solute to the perimeter. As the time t approaches
the drying time t
f
, the height h(r) decreases to zero as
(t
f
−t). But the outward current h(r)¯v(r) must stay con-
stant with time in order to replenish the constant evap-
oration flux J.Thus¯vmust grow as (t
f
− t)
−1
. This
diverging velocity leads to a diverging displacement of
any point r>0. Thus all initial points r>0 are carried
to the perimeter before the drying time t
f
.
100
1000
10000
M - M
o
10 100
t + t
o
1.37
10 100 1000
1000
10
100
FIG 3. Double-logarithmic plot of ring mass M(R, t)
versus time t for times from 10 to 250 seconds after
placement of the drop on the surface, for three differ-
ent drops whose total drying time was about 800 sec.
M(R, t) is measured in number of particles. Particle
number is counted in a sector of the ring similar to that
shown in Figure 1b. An offset M
0
= 2 was subtracted
from the lowest curve to account for early non-steady-
state deposition. M
0
= 50 and 40 for the middle and
upper curves. The upper curve coincided with the mid-
dle one and was shifted up by a factor of 2 for clarity.
An offset t
0
= 11 sec, 13 sec and 10 sec was added
to the time axis. The solid lines show the power law
(M −M
0
) = const (t+t
0
)
1.37
. Inset shows M(R, t) versus
t for the entire 1300-sec drying time. Circles: data ob-
tained from ten-micron spheres via counting. Roughly
90 percent of the particles were observed to go to the con-
tact line. Solid line: theoretical prediction determined
numerically.
To test these predictions we used drops of distilled
water which contained charge-stabilized surfactant-free
polystyrene microspheres [Interfacial Dynamics, Port-
land, Oregon] at a starting volume fraction of 10
−4
. The
spheres were so dilute that they could be regarded as
an ideal solution except in the very narrow ring region.
Droplets with nominal radius 4 mm were deposited on
glass microscope slides and allowed to dry in a large en-
closure with measured ambient temperature and humid-
2
ity. The volume of the droplet was inferred by weigh-
ing the slide during drying. The volume decreased at a
rate which agreed within two percent with that expected
for steady-state vapor-diffusion-limited evaporation, us-
ing tabulated values [10] for the water diffusivity in air
and for the saturated vapor concentration.
In a separate experiment we observed the solute ring
deposition by viewing the migration of one-micron micro-
spheres in a video microscope during drying, as shown in
Figure 1b. By automatically analyzing [11] the video
record, the depth-averaged velocity ¯v(r) and the number
of particles in the ring M(R, t) were measured. Contact
line pinning is produced by surface irregularities and is
much stronger with the solute than without it. The ring
deposit can create surface unevenness as well as augment
the surface imperfections that produced the initial pin-
ning.
The depth-averaged velocity ¯v(r) for thin drops, with
θ
c
' 0, shows the predicted (R − r)
−.5
divergence in
the vicinity of the contact line. The measured ring mass
M(R, t) is plotted in Figure 3. These drops had an initial
contact angle θ
c
' .25 radians. For this angle, our theory
predicts an initial increase M(R, t) ∼ t
1.37
. This behav-
ior is not expected at very early times (t
<
∼
(4R)
2
/D ' 10
sec), before steady-state diffusion has been achieved. To
account for transient effects during this early period, we
allow a shift in the effective starting time of order 10 sec.
We also include an offset in the deposited mass in order
to account for the particles deposited during the initial
transient. Choosing the M and t offsets to achieve the
best straight line on a log-log plot yielded power-law fits
M − M
0
∼ (t + t
0
)
p
, where p =1.3±0.1. Thus the ex-
pected initial growth of the ring is consistent with obser-
vation. To predict the growth of the ring at later times,
we first determine ¯v(r) numerically from the known flux
J(r) of a thick drop and then use this ¯v(r) to determine
M(R, t) as outlined after Eq. (1) above. This predicted
M(R, t) is compared with the data in the inset of Figure
3. Again the prediction is in good agreement with the
data.
Several effects modify the simplified theory sketched
above. Noncircular drops must have uneven deposition
rates: highly convex regions have a stronger evaporating
flux and thus denser deposits, as corroborated by Fig-
ure 1a. If the solute is not dilute, the ring deposit is
forced to have a nonzero width; higher initial concentra-
tion leads to a wider ring. Further thermodynamic ef-
fects may modify the flow and the ring deposition. Some
solutes may segregate to the solid surface and become
immobilized. Others may segregate to the free surface
where the outward flow is faster than ¯v. The thermal
and concentration gradients caused by evaporation can
lead to circulating surface-tension-gradient driven (i.e.,
Marangoni) flows. These can interfere with the outward
flow discussed above. Our experiments showed both sur-
face segregation and circulating flow but their impact was
minor (as we will discuss in detail elsewhere). High vis-
cosity in the liquid can also modify the deposition by pre-
venting the drop from attaining an equilibrium droplet
shape. We estimate that in our experiments such viscous
effects should be negligible except within a few microns
of the contact line.
Our measurements support this capillary flow mecha-
nism for contact-line deposition. They show that the de-
position can be predicted and controlled without knowing
the chemical nature of the liquid, solute or substrate. The
model accounts in a natural way for the nearly complete
transport of the solute to the periphery. By controlling
the speed and spatial variation of the evaporation, this
model predicts that we can control the shape and thick-
ness of the deposit. Often it is desired to deposit solute
particles in a confined region as in for example the print-
ing of fine lines [5]. The commonplace ring stain appears
to provide a simple and robust route to such confinement.
[1] Parisse, F. and Allain, C., ”Shape Changes of Colloidal
Suspension Droplets during Drying” J. Phys II, 6 1111-
1119 (1996).
[2] El Bediwi, A. B., Kulnis, W. J., Luo, Y., Woodland, D.,
and Unertl, W. N., ”Distributions of Latex Particles De-
posited from Water Suspensions” Mat. Res. S oc. Symp.
Proc., 372 277-282 (1995).
[3] Denkov, N. D., Velev, O. D., Kralchevsky, P. A., Ivanov,
I. B., Yoshimura, H., and Nagayama, K. ”Mechanism of
Formation of Two-Dimensional Crystals from Latex Par-
ticles on Substrates” Langmuir, 8 3183-3190 (1992).
[4] Laden, P. ed.Chemistry and technology of water based
inks (London, Blackie Academic & Professional, 1997)
[5] Society for Imaging Science and Technology, 1996 TAPPI
New Printing Technologies Symposium : proceedings (At-
lanta, GA : Tappi press,1996)
[6] Hisatake, K., Tanaka, S. Aizawa, Y., ”Evaporation of
Water in a Vessel” J. Appl. Phys. 73 7395-7401 (1993).
[7] Peiss, C. N., ”Evaporation of Small Water Drops Main-
tained at Constant Volume” J. Appl. Phys 655235-5237
(1989).
[8] Maxwell, J. C. Scientific Papers vol. 2 (Cambridge,
1890).
[9] Jackson, J. D.Classical Electrodynamics, 2nd edition (J.
Wiley, New York, 1975) p. 77.
[10] Lide, D. R. ed.CRC Handbook of Physics and Chem-
istry 77th edition, Chemical Rubber Publishing Com-
pany (Boca Raton FL, 1996). pp 6-218 6-8.
[11] Crocker, J. C. and Grier, D. G. ”Methods of Digital
Video Microscopy for Colloidal Studies” J. Colloid and
Interface Science, 179 298-310 (1996).
ACKNOWLEDGMENTS: The authors cordially thank
Hao Li, Xiangdong Shi and Michelle Baildon for their
3
early contributions to this project, John Crocker, David
Grier, and Andy Marcus for sharing with us their exper-
tise, their image analysis code, and their facilities, and
to Steve Garoff, L. Mahadevan, Sergei Esipov, Robert
Leheny, Dan Mueth, Ed Ehrichs, Jim Knight, Sean Blan-
ton, Narayanan Menon, Jeff Cina and Leo Kadanoff for
stimulating discussions. This work was supported by
grants from NSF-MRSEC, NSF, and DOE.
CORRESPONDENCE should be addressed to
RDD (rddeegan@control.uchicago.edu). Fur-
ther information may be found on the internet at
http://MRSEC.uchicago.edu/MRSEC
4
Ring deposition is a useful technology which can be used in printing, washing and coating processes. It might also be possible to stretch DNA using the strong shear flow that develops in ring forming drops when the evaporation rate is enhanced at the edge.
In 2011, researchers from the University of Pennsylvania showed that the coffee-drop effect can be negated if the particles are not spherical - the more elongated the particles, the more uniform the deposition. The shape of the particles is actually a way to control the distribution of material on a surface.
[Here's a video showing how deposition patterns depend on the shape of the particles.](https://www.youtube.com/watch?v=ZaCGoSTMHyc)
Here's a video by Robert Reegan on the effect
This paper ended up being cited more than 5000 times to this day. It's a fantastic example of an everyday effect with a simple explanation that remained a mystery until 1997.
### TL;DR
Robert Deegan is a professor of physics at the University of Michigan. In 1997, Deegan and his colleagues developed a theory to explain why a drying drop of coffee typically leaves behind a ring-shaped stain.
The ring is a consequence of the radially outward fluid flow induced by the so-called contact line pinning: the outer edge of a spilled coffee droplet grabs onto the solid surface and becomes pinned in place. The evaporating drop thus retains its pinned diameter and flattens while it dries.
"That flattening, in turn, is accompanied by fluid flowing from the middle of the drop toward its edge to replenish evaporating water. Suspended particles—the coffee grounds—are carried to the edge of the drop by that flow. Once there, they pile up, one at a time, into a tightly jammed packing and produce the coffee ring. Deegan and company studied the ring growth empirically by following the individual frames in a video of plastic colloidal spheres suspended in an evaporating droplet"
