Transcript Fadeout singularity of solute density in an evaporating drop --Rui Zhang, T.

Fadeout singularity of solute density in an evaporating drop
--Rui Zhang, T. Witten, University of Chicago http://jfi.uchicago.edu/~tten/Coffee.drops/
singularity
puzzle
background
solvent evaporates
What drives solute to the edge? Flow.
FADEOUT: solute density decreases smoothly with distance from the edge
solute --> ring
width of fadeout grows with solute concentration
J(x) t
Contact line is pinned
a means of strongly concentrating a solute in a controllable way
h
solute concentration 
h = J(x) t on average
h(x)
x
What controls the divergence of density as we approach the edge?
stain width and profile vary greatly, deterministically
converging to focus: v(x) increases
R
Height profile h(x) flattens uniformly: h = -hi t /T drying time
power law? log? Exponential?

But at edge, J(x) t >> h (0 )
how much can they vary?
universal? position dependent?
Thus fluid must flow towards edge to maintain constant shape: v(x)
lateral current: v(x) h(x)
how can they be controlled?
=
...later
.
1/4 (R-x)
uniform J
0
... v diverges as h0.
x
this v(x) carries every x to the boundary in time t < T
thus all solute
boundary
... strong focusing
profile
regimes
shock
Progress of shock front determines radial profile
Amount of solute matters ...NOT passive
passive solute concentration is determined by outward velocity v(x)
Dupont hypothesis:
Shock front L(t) motion determines deposit density
< (L, t)
Rui Zheng
Above c, solute immobilized...
solute stops flowing when
concentration  reaches c
less
solute
more
solute
... increases only via
decreasing height h(x)
flow
known
known
Dupont simulation shows realistic-looking
fadeout towards center
What is its functional form?
flux into
immobile
region
position

t
areal density
. ( /  )
c
+1
mobile
frozen
0
passive
solute
L(t)
- v flow converges to a stagnation point
position x
0.01
0.1
v
Single point controls fadeout everywhere!
fadeout:
(T-t)<<T
-7
log x
.01
-3
0.001
x (c/0)1/2
1.
xm~ 01/2
Poisson ratio at center governs fadeout power at edges
utility
Surprise! noncircular drops: single  controls fadeout
What good is evaporative deposition?

ri
Circular case
Consider distortion of a small disk of area A
during drying
h(0)
Combining, A
A
horizontal
strain
h
[
= - 1-
are in fixed ratio: e =
–
J(0) h
h
–
J h(0) h
2

p
–
J(0) h
–
J h(0)
h
vertical
strain
capillary drop
spherical cap
–
h = 1/2 h(0)
dry,
diffusioncontrolled
–
J = 2 J(0)
wet,
uniform
–
J = J(0)
3/8
1/4
gravity drop 1/4
–
flat h = h(0)
...like a solid.
Poisson ratio 
determined by simple averages
p=
Anisotropic case
0
Fadeout: central singularity dictates
tunable, global fadeout law.
Future
-1
with time
Further from equilibrium
r
Marangoni flows
viscous stresses
its area a (like cylinder area A) changes because
a h  a J t
v
reasoning for a is same as for A
Depinning patterns
A
except h has a new contribution due to translation
Fadeout profile is designable
3 
7
3
–
1 + 2 = 2 J h(0)
–
J(0) h
1 - 2
Dictated by smooth, deterministic dynamics.
controls fadeout at r
translates and distorts into
deposit
~ x-p
(x) ~
density
]

the area a
L(t) governs deposit density
• J controls h; total volume evaporated = volume removed
–
–
J
h(0)
h
(t) =
h(0)
average over drop
e ,
ri converges very anisotropically
Contraction of ri follows expansion of A: ri ~ h(0)
• h  e; flow is needed to make up the difference:
h - e = v = h(0) A
•
A~
Robust assembly of dilute reactants
Reproducible distribution in space
h-2
ri governs motion of the deposit front L(t): L ~ h(0)-1/2
• height decreases by h, reducing volume by h = (A h)
• Evaporation removes volume e = J(0) A (t)
How does ri approach stagnation point?
shapes


= -v( 
ri )
numerics
dry (diffusive J)
advance of
immobile
region
A/A = -2 h/h
d ri
dt
T
1/2
.1
fadeout
J(0)
grow? ie how fast does ri move?
wet (uniform J)
knowing when x freezes determines frozen  profile
Stagnation region deforms like compressible solid
how fast does
deposit
edge
stagnation
ri
dL ~ dm = amount of solute in
0.1 log density
< v h = (c-<)h dL/dt
areal
density
L, (T-t)
... maximum
<
Concentration must jump:

Time must be long after the max-deposition time––almost dry
deposit
density (L(t)) = c h(L, t) ~ L(t) * (T-t)
c
late deposition comes from stagnation point
Front equation (dilute)
R
x
Rui Zheng
at front, x = L(t)
v(L) = (c-<(L, t)) dL/dt
passive
concentration
known
concentration
more solute 
flow stops sooner, farther from edge
 wider deposit
source
1?
Happily this contribution is negligible near stagnation point
Thus, despite anisotropy a evolves just like A :
a
a
[
= - 1-
–
J(0) h
h
–
J h(0) h
]
2
There are two Poisson ratios. What happens to p ?
fadeout is governed by the same  as for isotropic case
area a
R. Deegan
U. C. PhD
thesis, 1998