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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) =