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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