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
ω
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
Quéré 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 hysteresisSchäffer 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. 2006-20084–7
(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 Hänggi (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
3η∇ [γ∆h + Π(h, r)] + µex
heterogeneous substrate (or field)
driving
z
x
pinning
liquid
Non-dimensional parameters
h . . . mean film thicknessLx × Ly . . . system size / periodµ . . . driving force
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 7
Periodic array of local wettability defect
Wettability via disjoining pressure
Π(h, r) = κ
(bh3 − [1 + εξ(x)] e−h
)ξ(x) = 2 cn[2K (k)x/L, k ]2 −∆
K (k) . . . complete elliptic integral of the first kind∆ . . . shift to have
∫ξ(x)dx = 0
Further parameters
ε . . . wettability contrastε < 0 hydrophilic defectε > 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
ε ξ(
x)
µa
0
1
2
3
h(x)
0.040.020.0050.0
10 20 30x
0
1
2
ε ξ(
x)
µb
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
µ0.2
0.4
0.6
0.8
1
1.2
||δh|
| 0.02 0.04µ0
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
ε0
0.01
0.02
0.03
µ
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
µ
0.4
0.6
0.8
1
||δh|
|
0.04 0.06 0.08µ
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
µ
||! h
||
0 2 4−2
−1.5
−1
−0.5
0
log(µ−µc)−const
log(
T)−c
onst
Steady drops
Time evolution
P. Beltrame, P. Hänggi, 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
µ
||δh||
µ3dsn2 µ3d
c1µ3dc2 µ3d
g
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
µ
rivulet
SSD
SSD
SR
SD
SSD
SR
SD
SR
SSR
0.05
0.1
Ph. Beltrame, E. Knobloch, P. Hänggiand 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
Partially wetting drop on rotating horizontal cylinder
Dimensional evolution equation in long-wave approximation
∂t h = −∂θ
h3 ∂θ [∂θθh + h − Bo cos(θ) + Π(h)] + Ωh,
θ
ω
g
h
R
Non-dimensional parameters
h = 1 . . . mean film thicknessL = 2π . . . system size / periodΩ = ηωR
ε3σ. . . Rotation number
Bo = R3ρghσ . . . Bond number
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 19
Drop families on resting horizontal cylinder (Ω = 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
||δh|
|
1.02.0
β0
(i)
(ii)
(iii)
0
1
2
3
h(θ)
0 3.14 6.28
θ
-1
0
1
cos
θ
(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 (Ω > 0)
L2-norm Profiles
0 1 2 3 4 5
Ω0
0.2
0.4
0.6
0.8
1
1.2
||δh|
|
0.50.751.02.55.010.0
Bo
0
1
2
3
h(θ)
0.00.51.01.68
0 3.14 6.28
θ
-1
0
1
cos
θ
Ω 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
Ω
0.94
0.96
0.98
1
1.02
||δh|
|
10-6
10-5
10-4
10-3
10-2
10-1
100
Ω−Ωsn
10-3
10-2
10-1
1/T
0π
2π 0
5
10
14.8
0123
t
θ
h
0π
2π 0
1
2
2.69
0123
t
θ
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
δF [h]
δh
with
F [h] =
∫V
[12
(∂xh)2 + f (h, x) + µxh]
dx
f (h, x) = −∫
Πdh . . . local free energy (heterogeneous system)µxh . . . potential energy for gravitational drivingQ(h) . . . mobility
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 23
ω
with A. Pototsky, A. J. Archer, S.E. Savel’ev, and F. MarchesoniA. Pototsky et al., PRE 83, 061401 (2011)
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 24
DDFT evolution equation as gradient dynamics
Evolution equation for conserved order parameter field ρ(x , t)
∂t ρ = ∂x
ρ(x , t) ∂x
δF [ρ(x , t)]
δρ(x , t)
with
F [ρ] = T∫ S
2
−S2
dx ρ[ln ρ− 1] +
∫ S2
−S2
dx Ueff(x)ρ
+ Fhc[ρ] + Fat[ρ]
Ueff(x) . . . external potentialFhc[ρ] . . . hard-core interactionFat[ρ] . . . attractive interaction
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 25
Contributions to energy
External potential
Ueff(x) = U(x)− Ax U(x) = sin (2πx) + 0.25 sin (4πx)
Attractive interaction
Fat[ρ] =12
∫ S2
− S2
dx∫ x+ S
2
x− S2
dx ′ wat(| x−x ′ |)ρ(x)ρ(x ′) with wat(x) = −αe−λx
Hard-core repulsion
Fhc[ρ] =12
∫ S2
− S2
dx φ[ρ(x)]
ρ
(x +
h2
)+ ρ
(x − h
2
)
where φ[ρ] = −T ln [1− η] and η(x , t) =∫ x+h/2
x−h/2 dx ′ ρ(x ′, t)
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 26
Numerical approach
Methods used for coupled 1d local ddft equation(s) [NOT HERE]• Path-following (steady solutions, saddle-node bifurcations)[AUTO07]• Time-integration [finite difference]
Methods used for nonlocal 1d ddft ratchet (integro-differentialequation)• Path-following of steady and time-periodic solutions[combining AUTO07 with FFT routine – Andrey Pototsky]
Wish list
−→ Path following in the 2d case for integro-differential equation−→ Adaptive grid
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 27
Depinning transition for 3L domain (with driving A = 1)
Bifurcation diagram
1 1.5 2 2.5 3
α
-0.4
-0.2
0
J
10-6.0
10-4.0
10-2.0
αhm
-α
0
50
100
150
τ
HB
LP
hm
τ=const ln(x)
α . . . interaction strength (∼ increasing contact angle)
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 28
Influence of size of particle core (h 6= 0)
0 2 4 6 8 10α
-0.9
-0.6
-0.3
0
J
4 6 8α
-3.5
-3
-2.5
-2
<F
/S>
-2 -1 0 1 2x
0
5
10
ρ(x
)3 3.2 3.4 3.6 3.8
α
-0.5
-0.25
J
3 3.2 3.4 3.6 3.8
α
-2
-1.8
<F
/S>
10-4.0
10-2.0
100.0
αhm
-α
0
100
200
300
τ
00.1 0.2
4L 4L 2L
4L
4L
4L
(e)
L
HB
BP
BP(a)
(b)
(c)
(d)
t-periodic
HBBP
BP
L
4L
HBBP
L
2L
4L
t-periodic
t-periodic
hm
hm
t-perio
dic
HB
(f)
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 29
Conclusions for depinning drops / probabilitydistributions
Depinning drops/ridges in 2d/3d
• Drops/ridges may depin via sniper/Hopf/homoclinic bifurcation• Stick-slip motion beyond depinning (sniper/homoclinic),related to translation mode• Intricate 3d behaviour (coupling to Plateau-Rayleighinstability)
DDFT for interacting particles in modulated nano-pore
• Interplay of steady-state (pitchfork) and depinning transitions(Hopf/homoclinic) [picture not yet complete]• Depinning dynamics via volume transfer or translation mode
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 30
Deposition patterns
with L. Frastia, A.J. ArcherUT et al., J Phys-Cond Mat 21, 264016 (2009);L. Frastia, A.J. Archer, UT, PRL 106, 077801 (2011);
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 31
Hydrodynamic equations in long-wave approximation I
φ(x)
h (x) h(x)
v(t)
z xp
Film thickness evolution equation (isothermal, without solute cf.Pismen 20028,9)
∂th = ∇ · [Q(h, φ)∇p(h)]−βρ
(p(h)− µρ)
Q(h, φ) =h3
3η(φ)mobility
p(h) = −γ∆h −Π(h) pressure
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 32
Hydrodynamic equations in long-wave approximation II
Evolution equation for effective ’layer thickness’ ψ = hφ
∂t(φh) = ∇ · [φQ(h, φ)∇p(h)] +∇ · [D(φ)h∇φ]
Viscosity/diffusivity of dense suspension [Quemada (1977)10,Krieger-Doherty law, Einstein relation)
η(φ) = η0
(1− φ
φc
)−ν
D(φ) = kBT/6πr0η(φ)
ν . . . Literature gives various exponents below ≈ 2
Evolution equations account for solvent capillarity, wettability,evaporation, solute diffusion, nonlinear rheology
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 33
Pattern types – Examples
Dried-in deposition patterns close to “first ring”
L. Frastia, A.J. Archer, UT, PRL 106, 077801 (2011)
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 34
Pattern types – morphological phase diagram
Depending on evaporation rate Ω and initial mean concentration φ0
(d)(b)(c)
(a)
(e)
(b)
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 35
Line patterns – dependence on concentration
Deposit profiles and characteristics for Ω0 = 4.64×10−7
x
low
medium
high
Transition to line deposition as depinning transition in comovingframe→ depinning via Hopf and infinite period bifurcation
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 36
LANGMUIR-BLODGETT TRANSFER
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 37
Patterning by Langmuir-Blodgett transfer
Chi, Fuchs et al (2004) – Transfer of DPPC onto silicon oxide
SFM images, bar 2µm
see also Riegler et al 1992/94
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 38
Thin film model by Köpf, Gurevich, Friedrich, and Chi(2010)
Geometry
Evolution equations
∂t Γ = −∇ ·[
Γh2
2∇p + Γh∇Σ− V Γ
]∂th = −∇ ·
[h3
3∇p +
h2
2∇Σ− Vh
]− Ωµ(h, Γ)
Resulting line patterns
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 39
Efficient description with Cahn-Hilliard-type model
Depinning bifurcation diagram
Homoclinic (low V) and Hopf (high V) bifurcationSnaking of localised (front) states
Köpf, Gurevich, Friedrich, UT (submitted 2011)
Steadyprofiles
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 40
Numerical approach
Methods used for 1d Langmuir Blodgett transfer• Path-following (steady solutions) [AUTO07]• Time-integration [finite difference]
Problem• Did not obtain agreement in sufficient detail to establishwhere the time-dependent branch emerges from
Solution• Coding of path-continuation and time-integration employingexactly the same spatial discretization (Michael Köpf)
Wish list
−→ Path-following and time-integration in one package
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 41
General conclusions
Depinning in many driven heterogeneous soft-matter systems
• Drops/ridges on heterogeneous substrates• Clusters of interacting particles in heterogeneous nanopores• Deposition of solutes from contact lines• Langmuir-Blodget transfer of surfactant-layers
Depinning occurs via Hopf, sniper or homoclinic bifurcations
• Typical scaling of period close to transition• Involves translational and/or volume mode
Heterogeneity might be
• Due to imposed modulations throughout the domain• Due to boundary effects• Self-organised (not discussed, e.g., PFC)
Relation to snaking ?Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 42
References I
[1] D. Quéré, M. J. Azzopardi, and L. Delattre. Drops at rest on a tilted plane. Langmuir, 14:2213–2216, 1998.
[2] E. B. Dussan. On the spreading of liquids on solid surfaces: Static and dynamic contact lines. Ann. Rev. FluidMech., 11:371–400, 1979. doi: 10.1146/annurev.fl.11.010179.002103.
[3] E. Schäffer and P. Z. Wong. Dynamics of contact line pinning in capillary rise and fall. Phys. Rev. Lett., 80:3069–3072, 1998. doi: 10.1103/PhysRevLett.80.3069.
[4] J. Xu, J. F. Xia, S. W. Hong, Z. Q. Lin, F. Qiu, and Y. L. Yang. Self-assembly of gradient concentric rings viasolvent evaporation from a capillary bridge. Phys. Rev. Lett., 96:066104, 2006. doi:10.1103/PhysRevLett.96.066104.
[5] S. W. Hong, J. F. Xia, and Z. Q. Lin. Spontaneous formation of mesoscale polymer patterns in an evaporatingbound solution. Adv. Mater., 19:1413–1417, 2007. doi: 10.1002/adma.200601882.
[6] J. Xu, J. F. Xia, and Z. Q. Lin. Evaporation-induced self-assembly of nanoparticles from a sphere-on-flatgeometry. Angew. Chem.-Int. Edit., 46:1860–1863, 2007. doi: 10.1002/anie.200604540.
[7] S. W. Hong, W. Jeong, H. Ko, M. R. Kessler, V. V. Tsukruk, and Z. Q. Lin. Directed self-assembly of gradientconcentric carbon nanotube rings. Adv. Funct. Mater., 18:2114–2122, 2008. doi: 10.1002/adfm.200800135.
[8] A. V. Lyushnin, A. A. Golovin, and L. M. Pismen. Fingering instability of thin evaporating liquid films. Phys.Rev. E, 65:021602, 2002. doi: 10.1103/PhysRevE.65.021602.
[9] L. M. Pismen. Mesoscopic hydrodynamics of contact line motion. Colloid Surf. A-Physicochem. Eng. Asp.,206:11–30, 2002.
[10] D. Quemada. Rheology of concentrated disperse systems and minimum energy-dissipation principle I.Viscosity-concentration relationship. Rheol. Acta, 16:82–94, 1977.
Uwe Thiele, Loughborough University – www.uwethiele.de Leiden, November 2011 43
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