Variational Principle (Onsager Principle) in Soft Matter Dynamics · PDF file...

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Transcript of Variational Principle (Onsager Principle) in Soft Matter Dynamics · PDF file...

  • Masao Doi Center of Soft Matter Physics and its

    Applications, Beihang University, Beijing China

    2017/01/09 Santiago Chilie

    Variational Principle (Onsager Principle) in Soft Matter Dynamics

  • Dissipative Lagrangian mechanics

    i

    d L L 0 dt x x x

     ∂ ∂ ∂Φ   − + =    ∂ ∂ ∂      

    21 x 2

    Φ = ζ  Dissipation function

    mx kx x 0+ + ζ = 

    2 21 1L K U mx kx 2 2

    = − = − Lagrangean

    The equation is not validated. Mass renormalization is ignored.

  • Overdamped limit: frictional force >> inertial force

    U 0 x x

    ∂ ∂Φ   + =   ∂ ∂    U x 0 x

    ∂ + ζ =

    ∂ 

    L K U U= − ≈ −

    2

    R(x, x) U 1 Ux x 2 x

    = Φ + ∂

    = ζ + ∂

     

    Rayleighian

    R 0 x

    ∂ =

    ∂

    21 x 2

    Φ = ζ 2 1U kx 2

    =

    Time evolution of the system is given by

  • Hydrodynamic variational principle

    ij i j i i

    1 UR[x, x] (x)x x x 2 x

    ∂ = ζ +

    ∂∑ ∑   

    Particle motion is determined by minimizing

    2 vvR[v(r); x, x] d U

    4 r r βα

    β α

     ∂∂η = + +  ∂ ∂ 

    ∫ r  Minimize with respect to v(r)

    ij j i

    U(x)x 0 x ∂

    ζ + = ∂∑ 

    Minimize with respect to ix

    )x,...x,x(x f21= particle coordinates U(x) potential energy

    jiij ζ=ζ

  • HF

    xV H F

    yV

    HFωHTV

    Reciprocal relation is not trivial

    jiij ζ=ζ

    yxxy ζ=ζ

    xrrx ζ=ζ

    Hy Hx

    x y

    F F V V

    =

    H HT F V

    = ω

  • Onsager principle

    )x,...x,x(x f21=

    A(x)

    State variables specifying the non-equilibrium state

    Free energy

    ij i j i i

    1 AR(x; x) (x)x x x 2 x

    ∂ = ζ +

    ∂∑ ∑   

    Time evolution is given by

    Dissipation function Free energy change rate

    ij j i

    Ax 0 x ∂

    − ζ − = ∂∑ 

    )x()x( jiij ζ=ζOnsager’s reciprocal relation

    Time evolution is given by which minimisesix

  • Applications

  • Meniscus Rise

    21 hh 2

    Φ = η Dissipation function

    )t(h

    State variable

    2 21U a gh 2 ah 2

    = ρπ − γ π

    )t(h

    Free energy

  • Wetting of dry sand 2

    2 21R hh gha h ah 2

    = η +ρ − γ  

    2hh gha a 0η +ρ − γ =

    t

    )t(h

    eqh ga γ

    ≈ ρ

    ah(t) tγ≈ η

    )t(h

  • Diffusion

    State variable )t;x(n Free energy )]x(nln)x(nTk[dx)]x(n[A B∫=

    )]x(n[ Φ

    )nv( x

    n p∂ ∂

    −=

    ∫ ζ=Φ 2pnvdx2 1

    It is difficult to write the dissipation function in the form of

    But cab be written as )x(n

    Then

  • Diffusion 2 )]x(nln)x(nTk[dx)]x(n[A B∫=

    )nv( x

    n p∂ ∂

    −=

    pBpB

    pB

    v x ndxTknv

    x ]1n[lndxTk

    )nv( x

    ]1n[lndxTk

    )x(n]1)x(n[lndx)]x(n[A

    ∫∫

    ∫ ∫

    ∂ ∂

    = ∂ +∂

    =

     

     

    ∂ ∂

    −+=

    += 

    x nvdxTknvdx

    2 1R pB

    2 p ∂

    ∂ +ζ= ∫∫

    ζ =

    TkD B2 2

    x nD

    t n

    ∂ ∂

    = ∂ ∂

    0 x nTknv Bp =∂ ∂

  • Diffusion equation is an Onsager’s kinetic equation

    2

    2

    x nD

    t n

    ∂ ∂

    = ∂ ∂

    )]x(nln)x(nTk[dx)]x(n[A B∫=

    )y(n A)y,x(dy

    t n

    δ δ

    µ−= ∂ ∂

    is written as

    )x,y()y,x( µ=µ

  • Many transport equations known in soft matter can be derived from the Onsager principle

    • Stokes equations • Diffusion equations • Smoluchowskii equation • Cahn-Hilliard equation in phase separation • Ericksen-Leslie equation in liquid crystals • Gel dynamic equation • …..

    see Soft matter physics (OUP 2013)

  • Onsager principle as a tool of approximation

  • Searching the next state at

    R(x; x) (x; x) A(x; x)= Φ +   

    The evolution of the state x=(x1,x2,…) is given by the minimum of

    If the current state is x, the state at the next time step is given by the minimizng R( x / t; x)∆ ∆

    We search the minimum in a subset of nonequilibrium states

    i ix x ( )= α

    ij i j i i

    1 UR(x; x) (x)x x x 2 x

    ∂ = ζ +

    ∂∑ ∑    21 AR(x; x)

    2 α ∂

    = ζ α + α ∂α

     

    t t+ ∆

  • An approximate calculation using Onsager principle

    1 xn(x, t) |1 | | x | a a(t) a(t)

    = − <

    p av x a

    = 

    a− x

    )t,x(n

    a

    2 2 B p B p

    2k T1 n 1R dx nv k T dx v a a 2 x 6 a

    ∂ = ζ + = ζ −

    ∂∫ ∫  

    )nv( x

    n p∂ ∂

    −=

    B2k T1 a 3 a ζ =

    Assume

    aa 6D= a(t) 12Dt=

  • Comparison

    21 xn(x, t) exp 4Dt4 Dt

      = − π  

    1 xn(x, t) |1 | 12Dt 12Dt

    = −

    0 2-2 0

    2 Dt=0.01

    0.1 0.5

  • Sliding droplet on a slope

    α

  • 3D problem

    1 2 3 4H(x, t) (x a )(a x)(a a x)= − − +

    1 6a (t),...a (t)

    x

    z x

    y

    2 yz h(x, y, t) H(x, t) 1

    Y(x, t)

       = = −  

       

    1/2 1/2 1 2 5 6Y(x, t) (x a ) (a x) (a a x)= − − +

    are determined by the Onsager principle

  • Shapes in steadily slidinjg droplets

    Xianmin Xu, Yana Di, MD, Phys, Fluid (2016)

  • Meniscus rise between a flexible sheet and a rigid wall

    T. Cambau, J. Bico and E Reyssat EPL (2011)

  • Meniscus rise between a flexible sheet and a rigid wall

    h

    2wm 2w

    em 2w

    e hm

    e

    2 2 m

    m3 m

    (e e)1 3A 2 hw gewh B h 2 2 (w w)

    − = − γ + ρ +

    − h

    2 z

    0

    w 12dz v 2 e

    η Φ = ∫

    z ev h (h z) e

    = − − 

     zeve

    z ∂

    = − ∂

  • Meniscus rise between a flexible sheet and a rigid wall

    2/7

    0 0 2

    0

    e Ut7h(t) h 1 3 h

      = + 

     

    3/7

    0 0 2

    0

    e Ut7e(t) e 1 3 h

    −  

    = +   

    Di Y, Xu X. MD EPL (2016)

    U γ= γ

  • Deposit pattern in drying droplet

    Coffee ring

    When contact line is not pinned

    When contact line is pinned

    Coffee ring Mountain Volcano

    Xingkun Man, MD PRL (2016)

  • Swelling of a gel

    a2

    b2

    c2 In weak swelling, the aspect ratio of the gel remains almost constant

    a(t) : b(t) : c(t)

  • Kinetics in spray drying

    F. Meng et al EPJE (2015)

  • The evolution of the state x=(x1,x2,…) is given by the minimum of

    Approximate solution is obtained by the minimum in a restricted space i ix x ( )= α

    ij i j i i

    1 AR(x; x) (x)x x x 2 x

    ∂ = ζ +

    ∂∑ ∑   

    21 AR(x; x) 2 α

    ∂ = ζ α + α

    ∂α  

    Summary and Unsolved Problems

    • Is there any criterion which tells us which approximation is the best?

    • Does Onsager principle pose any general properties for the solution?

    Unsolved problems

    Summary