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Case study in continuation:Depinning transitions in selected driveninterface-dominated soft-matter systems
Uwe Thiele
Leiden, November 2011
ITN MULTIFLOW
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 1
Outline
Depinning drops
g
Drop on rotating cylinder
Interacting driven particlesin a nano-pore
Conclusions and Outlook
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 2
Experiment: Drops resting on a tilted plane
Qur et al. (1998)1
Geometry Depinning threshold
Determine the shape of the pinned drop, and the depinningthreshold?
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 3
Contact line pinning, contact angle hysteresis andstick-slip motion
Dussan (1979)2
Contact angle hysteresisSchffer and Wong (1998)3
Stick-slip motion
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 4
Dewetting evaporating suspensions line patterns
Zhiqun Lin et al. 2006-200847
(CdSe/ZnS core/shell, 4.4nm (right) and 5.5nm (left) in diameter, in toluene)
Concentrations: 0.25mg/ml, 0.15mg/ml, 0.05mg/ml,
0.05mg/ml
Concentration: 0.25mg/ml
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 5
with E Knobloch (2d/3d), P Beltrame (3d), P Hnggi (3d)
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 6
Driven drops/film on heterogeneous substrate
Dimensional evolution equation in long-wave approximation
t h = {
h3
3 [h + (h, r)] + ex
}
heterogeneous substrate (or field)
driving
z
x
pinning
liquid
Non-dimensional parameters
h . . . mean film thicknessLx Ly . . . system size / period . . . driving force
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 7
Periodic array of local wettability defect
Wettability via disjoining pressure
(h, r) = (
bh3 [1 + (x)] eh
)(x) = 2 {cn[2K (k)x/L, k ]}2
K (k) . . . complete elliptic integral of the first kind . . . shift to have
(x)dx = 0
Further parameters
. . . wettability contrast < 0 hydrophilic defect > 0 hydrophobic defect
s log(1 k) . . . steepness of defect
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 8
Numerical approach
Methods used in 2d (1d equation) Path-following (steady solutions, saddle-node bifurcations)[AUTO2000/AUTO07] Time-integration [finite difference, adaptive time-step for stiffequations - NAG]
Methods used in 3d (2d equation) Path-following (steady solutions) Time-integration [exponential propagation, adaptive time-step](Both algorithms employ a Cayley-Arnoldi method)Philippe Beltrame and UT, SIADS 9, 484 (2010)
Wish list
Path following for time-periodic solutions (2d and 3d) Adaptive grid for 2d [AUTO2014 for 4th (and 6th) order PDEs]
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 9
Profiles of pinned driven 2d drops (3d transversallyinvariant ridges
Hydrophilic defect Hydrophobic defect
0
1
2
3
h(x)
0.01780.010.0050.0
10 20 30x
-2
-1
0
(
x)
a
0
1
2
3
h(x)
0.040.020.0050.0
10 20 30x
0
1
2
(
x)
b
UT and E. Knobloch: PRL 97, 204501 (2006); NJP 8, 313 (2006)
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 10
Dynamics of depinning of a 2d drop pinned by ahydrophilic defect
Depinning via sniper bifurcation Stick-slip infinitely slow attransition.
0 0.02 0.04
0.2
0.4
0.6
0.8
1
1.2
||h|
| 0.02 0.040
0.01
T-1
a
0
25 0
100.7
0
4
t
x
h
0
25 0
740.7
0
4
t
x
h
b
c
UT and E. Knobloch: PRL 97, 204501 (2006); NJP 8, 313 (2006)
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 11
Pinning/depinning phase diagram
-0.8 -0.4 0 0.4 0.8
0
0.01
0.02
0.03
depinneddrops
pinneddrops
pinneddrops
hydrophilicdefect
hydrophobicdefect
a
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 12
Depinning via Hopf bifurcation (hydrophobic case, 2d)
Stick-slip with finite velocity at transition
0 0.02 0.04 0.06 0.08
0.4
0.6
0.8
1
||h|
|
0.04 0.06 0.08
0
0.01
0.02
T-1
a
0
25 0
47.4
0
4
t
x
h
0
25 0
206.4
0
4
t
x
h
b
c
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 13
3d drop - hydrophilic line defect (sniper)
Bifurcation diagram
0 0.005 0.01 0.0150.45
0.5
0.55
0.6
0.65
0.7
0.75
||! h
||
0 2 42
1.5
1
0.5
0
log(c)const
log(
T)c
onst
Steady drops
Time evolution
P. Beltrame, P. Hnggi, UT; EPL 86, 24006 (2009)
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 14
Multistability in 3d?
Full picture has to relate
Depinning of 3d drops and transversally invariant ridges(2d drops)Plateau-Rayleigh instability of a ridge (zero/finite drivingwith/without heterogeneityRivulet solutions and their stability w.r.t. surface waveswith/without heterogeneity
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 15
Bifurcation scenario medium lateral system size
0 0.002 0.004 0.006 0.008 0.01 0.012
0.6
0.65
0.7
0.75
0.8
0.85
0.9
||h||
3dsn2 3dc1
3dc2
3dg
SR1
+
+++
++
+++
SD1
+
SD2
+
SD3++
SSD
SSR
SR2++
3d drops annihilate in saddle-node (no sniper bifurcation)
Depinning spanwise invariant ridges are unstable from onset
Stick-slipping drops result from (subcritical) homoclinicbifurcation
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 16
Stability diagram for steady and stick-slipping states
20 22 24 26 28 30 32
L
0
0.005
0.01
rivulet
SSD
SSD
SR
SD
SSD
SR
SD
SR
SSR
0.05
0.1
Ph. Beltrame, E. Knobloch, P. Hnggiand UT, PRE 83, 016305 (2011)
Black solid (dashed):transversal (sniper)instability of steadyridges
Thick dotted blue(dashed red):saddle-node (sniper)bifurcations of thedrop states
Thin dotted blue:hypothetical border ofthe region of stablestick-slipping drops
Dot-dashed green:Hopf bifurcation wherethe steady 3d rivuletsbecome stable
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 17
g
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 18
Partially wetting drop on rotating horizontal cylinder
Dimensional evolution equation in long-wave approximation
t h = {
h3 [h + h Bo cos() + (h)] + h},
g
h
R
Non-dimensional parameters
h = 1 . . . mean film thicknessL = 2 . . . system size / period = R
3. . . Rotation number
Bo = R3gh . . . Bond number
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 19
Drop families on resting horizontal cylinder ( = 0)
L2-norm Profiles
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Bo0
0.2
0.4
0.6
0.8
1
||h|
|
1.02.0
0
(i)
(ii)
(iii)
0
1
2
3
h()
0 3.14 6.28
-1
0
1
cos
(i)
(ii)
(iii)
a
b
-10 0 10x
-10
0
10
y
c
UT, JFM 671, 121-136 (2011)
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 20
Drop families on rotating horizontal cylinder ( > 0)
L2-norm Profiles
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
1.2
||h|
|
0.50.751.02.55.010.0
Bo
0
1
2
3
h()
0.00.51.01.68
0 3.14 6.28
-1
0
1
cos
a
b
-10 0 10x
-10
0
10
y
c
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 21
Rotating cylinder depinning transition via sniper
L2-norm / period Space-time plots
1 1.5 2 2.5
0.94
0.96
0.98
1
1.02
||h|
|
10-6
10-5
10-4
10-3
10-2
10-1
100
sn
10-3
10-2
10-1
1/T
0
2 0
5
10
14.8
0123
t
h
0
2 0
1
2
2.69
0123
t
h
Uwe Thiele, Loughborough University www.uwethiele.de Leiden, November 2011 22
Thin film equation as gradient dynamics
Evolution equation for conserved order parameter field
t h = x
{Q(h) x
F [h]h
}with
F [h] =
V
[12
(xh)2 + f (h, x) + xh]
dx
f (h, x) =
