Case study in continuation: Depinning transitions in ... U... · Case study in continuation:...

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


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?


Contact line pinning, contact angle hysteresis andstick-slip motion

Dussan (1979)2

Contact angle hysteresisSchäffer and Wong (1998)3

Stick-slip motion


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


with E Knobloch (2d/3d), P Beltrame (3d), P Hänggi (3d)


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


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


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]


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)


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)


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


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


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)


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


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


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


g


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


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)


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


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


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


ω

with A. Pototsky, A. J. Archer, S.E. Savel’ev, and F. MarchesoniA. Pototsky et al., PRE 83, 061401 (2011)


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


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)


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


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)


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)


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


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);


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


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


Pattern types – Examples

Dried-in deposition patterns close to “first ring”

L. Frastia, A.J. Archer, UT, PRL 106, 077801 (2011)


Pattern types – morphological phase diagram

Depending on evaporation rate Ω and initial mean concentration φ0

(d)(b)(c)

(a)

(e)

(b)


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


LANGMUIR-BLODGETT TRANSFER


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


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


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


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


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.


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