Onsager Principle - gateway.newton.ac.uk · Diffusion equation is an Onsager’s kinetic equation 2...
Transcript of Onsager Principle - gateway.newton.ac.uk · Diffusion equation is an Onsager’s kinetic equation 2...
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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 -
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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
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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
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How it startedIn chapter 3 of “Theory of polymer dynamics”, we needed to derived evolution equations for general polymer models
John Kirkwood 195?
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Variational principle for polymer dynamics
Sam said, K should be called “Rayleighian”.
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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)
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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
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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
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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
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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
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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
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Onsager principle as a tool of approximation
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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+ ∆
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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=
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Comparison
21 xn(x, t) exp4Dt4 Dt
= − π
1 xn(x, t) |1 |12Dt 12Dt
= −
0 2-20
2Dt=0.01
0.10.5
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Examples
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1. Droplet motion on a slope
α
Xianmin Xu, Yana Di, MD, Phys, Fluid (2016)U
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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
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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
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Receding contact angle in steady state
*U / U
advθ
recθ
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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
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Shapes in steadily slidinjg droplets
Xianmin Xu, Yana Di, MD, Phys, Fluid (2016)
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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)
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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∂
= −∂
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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
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Change of deposition pattern
Large CLξ Small CLξ
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Protein suspensions (milk)
Silica colloids
Spray drying
3. Drying of milk droplet
Sadek et al Langmuir 2013
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Mechanism of cavity formation
An elastic layer is created at the surface
F. Meng MD, Z. OuYang PRL (2014)
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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)
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Gelation and Cavitation
F. Meng et al EPJE (2015)
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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