Post on 11-May-2020
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