Masao Doi Center of Soft Matter Physics and its

Applications, Beihang University, Beijing China

2016/09/09 Soft Matter- Theoretical and Industrial Challenges -

Onsager Principle- A Useful Principle in Solving Industrial

Problems -

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 ij ji

Ax 0x∂

− ζ − =∂∑

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

Time evolution is given by which minimisesix

Lars Onsager Phys. Rev 1931

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 ij ji

Ax 0x∂

− ζ − =∂∑

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

Time evolution is given by which minimisesix

Lars Onsager Phys. Rev 1931

V

U= −∇F

viscous potential

How it startedIn chapter 3 of “Theory of polymer dynamics”, we needed to derived evolution equations for general polymer models

John Kirkwood 195?

Variational principle for polymer dynamics

Sam said, K should be called “Rayleighian”.

Dissipative Lagrangian mechanics

i i

U 0x x

∂ ∂Φ− − = ∂ ∂

Over damped limit, the kinetic equation is written as

i i i

d L L 0dt x x x ∂ ∂ ∂Φ

+ − = ∂ ∂ ∂

∑=Φ jiij xxζ21

Dissipation fn

Uxx

∂ζ = −

∂

ij i j ii

1 UR(x; x) (x)x x x2 x

∂= ζ +

∂∑ ∑

Minimize Rayleighian R(x; x)

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 which minimisesix

ij ji

Ax 0x∂

− ζ − =∂∑ Minimize R(x; x)

Dissipation function Free energy change rate

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, Masao Doi, OUP 8

A simple application : 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∫=

ijij x

Ax)x(ζ∂∂

−=∑ ∑ ∂∂

µ−=j

iji xA)x(x

Onsager’s kinetic equation

)y(nA)y,x(dy

tn

δδ

µ−=∂∂

∫is written as

)x,y()y,x( µ=µ

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

Examples

1. Droplet motion on a slope

α

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

α= singgx

0(t )(x x (t ))eh(x, t) h 1 e−κ − = −

At equilibrium Moving at velocity U

g

e 0(x x )eq eh (x) h 1 e−κ − = −

ηρ

=3

hgU2ex

g

At equilibrium

γρ

=κg

e

eee hκ=θ

When receding

Contact angle

e(t) (t)hθ = κ

Analysis for 2D case

Evolution equation for the receding contact angle

)xx)((xhggh21'h

21dxA 01SLSVx

22x

x

1

0

−γ−γ−γ+

ρ+ρ+γ= ∫

2x

x

vh

3dx21 1

0

η=Φ ∫

0

x

x

1v(x) dx ' h(x ')h(x)

= − ∫

*U γ=η

Receding contact angle in steady state

*U / U

advθ

recθ

3D calculation

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)

2. Deposition pattern on substrate

Coffee ring

When contact line is not pinned

When contact line is pinned

Coffee ring Mountain Volcano

Xingkun Man, MD PRL (2016)

Simple case:Wetting of a Newtonian droplet

2

2rh(r, t) H(t) 1

R(t)

= −

R2

0

1 3dr2 r v2 h

ηΦ = π∫

R(t)

H(t)

surf wetA A A= +

Assume

2HR const=

State variable R(t)

r

h(r, t)

( A) 0R∂

Φ + =∂

1/10R(t) t∝ Tanner’s law

1 (hvr)hr r∂

= −∂

Deposition of particles in an evaporating droplet

R2 2

CL0

1 3 1dr2 r v R2 h 2

ηΦ = π + ξ∫

J: evaporation rate (assumed to be constant)

Particles move with fluid velocity v(r)

1 (hvr)h Jr r∂

= − −∂

Volume conservation includes the effect of evaporation

2

2rh(r, t) H(t) 1

R(t)

= −

Assume

Contact line moves with velocity R

( A) 0R∂

Φ + =∂

Contact line friction(assumed to be constant)

CLξ

R, v(R) Deposition position

Change of deposition pattern

Large CLξ Small CLξ

Protein suspensions (milk)

Silica colloids

Spray drying

3. Drying of milk droplet

Sadek et al Langmuir 2013

Mechanism of cavity formation

An elastic layer is created at the surface

F. Meng MD, Z. OuYang PRL (2014)

Solute based Lagrangian scheme

2

2nn D

x∂

=∂

Drying of a film of colloidal solution

The particles located at x at time 0 moves to X(x,t) at time t

2X"X DX '

=

Euler picture: how particle concentration at x changes in time

Lagrange picture: how particles at x move in time

Advantage: gelation is easily handled

F. Meng et al EPJE (2015)

Gelation and Cavitation

F. Meng et al EPJE (2015)

Conclusion•The evolution law in soft matter (fluid motion, diffusion, phase change) can be stated in the form of the Onsager principle.•This gives us a new way of looking at problems, and a new way of solving the problems.•It has been applied to problems where rheologyis coupled with diffusion and phase changes (evaporation, structural formation)•Attempts are made to generalize the method to include

–viscoelasticity–contact line hysteresis