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 0dt x x x

∂ ∂ ∂Φ − + = ∂ ∂ ∂

21 x2

Φ = ζ Dissipation function

mx kx x 0+ + ζ =

2 21 1L K U mx kx2 2

= − = − Lagrangean

The equation is not validated. Mass renormalization is ignored.

Overdamped limit:frictional force >> inertial force

U 0x x

∂ ∂Φ + = ∂ ∂ U x 0x

∂+ ζ =

∂

L K U U= − ≈ −

2

R(x, x) U1 Ux x2 x

= Φ +∂

= ζ +∂

Rayleighian

R 0x

∂=

∂

21 x2

Φ = ζ 21U kx2

=

Time evolution of the system is given by

Hydrodynamic variational principle

ij i j ii

1 UR[x, x] (x)x x x2 x

∂= ζ +

∂∑ ∑

Particle motion is determined by minimizing

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

4 r rβα

β α

∂∂η= + + ∂ ∂

∫ r

Minimize with respect to v(r)

ij ji

U(x)x 0x∂

ζ + =∂∑

Minimize with respect to ix

)x,...x,x(x f21= particle coordinatesU(x) potential energy

jiij ζ=ζ

HF

xV HF

yV

HFωHTV

Reciprocal relation is not trivial

jiij ζ=ζ

yxxy ζ=ζ

xrrx ζ=ζ

Hy Hx

x y

F FV V

=

H HT FV

=ω

Onsager principle

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

A(x)

State variables specifying the non-equilibrium state

Free energy

ij i j ii

1 AR(x; x) (x)x x x2 x

∂= ζ +

∂∑ ∑

Time evolution is given by

Dissipation function Free energy change rate

ij ji

Ax 0x∂

− ζ − =∂∑

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

Time evolution is given by which minimisesix

Applications

Meniscus Rise

21 hh2

Φ = η Dissipation function

)t(h

State variable

2 21U a gh 2 ah2

= ρπ − γ π

)t(h

Free energy

Wetting of dry sand 2

2 21R hh gha h ah2

= η +ρ − γ

2hh gha a 0η +ρ − γ =

t

)t(h

eqhgaγ

≈ρ

ah(t) tγ≈

η

)t(h

Diffusion

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

)]x(n[ Φ

)nv(x

n p∂∂

−=

∫ ζ=Φ 2pnvdx

21

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

vxndxTknv

x]1n[lndxTk

)nv(x

]1n[lndxTk

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

∫∫

∫∫

∂∂

=∂+∂

=

∂∂

−+=

+=

xnvdxTknvdx

21R pB

2p ∂

∂+ζ= ∫∫

ζ=

TkD B2

2

xnD

tn

∂∂

=∂∂

0xnTknv Bp =∂∂

+ζ

Diffusion equation is an Onsager’s kinetic equation

2

2

xnD

tn

∂∂

=∂∂

)]x(nln)x(nTk[dx)]x(n[A B∫=

)y(nA)y,x(dy

tn

δδ

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 ii

1 UR(x; x) (x)x x x2 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 | aa(t) a(t)

= − <

pav xa

=

a− x

)t,x(n

a

2 2 Bp B p

2k T1 n 1R dx nv k T dx v a a2 x 6 a

∂= ζ + = ζ −

∂∫ ∫

)nv(x

n p∂∂

−=

B2k T1 a3 aζ =

Assume

aa 6D=

a(t) 12Dt=

Comparison

21 xn(x, t) exp4Dt4 Dt

= − π

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

= −

0 2-20

2Dt=0.01

0.10.5

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

zx

y

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

Y(x, t)

= = −

1/2 1/21 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

2we

hm

e

22 m

m3m

(e e)1 3A 2 hw gewh B h2 2 (w w)

−= − γ + ρ +

−h

2z

0

w 12dz v2 e

ηΦ = ∫

zev h (h z)e

= − −

zeve

z∂

= −∂

Meniscus rise between a flexible sheet and a rigid wall

2/7

00 2

0

e Ut7h(t) h 13 h

= +

3/7

00 2

0

e Ut7e(t) e 13 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 ii

1 AR(x; x) (x)x x x2 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