Supercooled liquids
Zhigang Suo
Harvard University
Prager Medal Symposium in honor of Bob McMeeking
SES Conference, Purdue University, 1 October 2014
1
Mechanics of supercooled liquids
Jianguo Li
Qihan Liu
Laurence Brassart
Journal of Applied Mechanics 81, 111007 (2014)
2
Supercooled liquid
supercooled
liquid
melting point
Volume
liquid
crystal
Temperature
3
A simple picture of liquid
• A single rate-limiting step: molecules change neighbors
• Two types of experiments: viscous flow and self-diffusion
4
Stokes-Einstein relation
hD
1
=
kT Sb
particle
b
liquid
h
f,v
f
v=
Sbh
Stokes (1851)
D
v=
f
kT
Einstein (1905)
5
Success and failure of Stokes-Einstein relation
TNB
IMC
OTP
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014). Based on experimental data in the literature
6
A supercooled liquid forms a dynamic structure
The dynamic structure jams viscous flow, but not self-diffusion.
Ediger, Annual Review of Physical Chemistry 51, 99 (2000).
7
Our paper
Given that the Stokes-Einstein relation fails,
we regard viscous flow and self-diffusion as independent processes,
and formulate a “new” fluid mechanics.
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
Homogeneous state
Helmholtz free energy
of a composite system
0-V s ij dij -V m R £ 0
Liquid
force
reservoir
Incompressible molecules
dkk = WR
æ
m ö
çs ij + dij ÷ dij ³ 0
W ø
è
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
9
Thermodynamic equilibrium
æ
m ö
çs ij + dij ÷ dij ³ 0
W ø
è
reservoir
liquid
membrane
m
s ij + dij = 0
W
s 12 = s 23 = s 31 = 0
m
s 11 = s 22 = s 33 = - dij
W
osmosis
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
10
Linear, isotropic, viscous, “porous” liquid
æ
m ö
çs ij + dij ÷ dij ³ 0
W ø
è
æ
ö
m
1
s ij + dij = 2h ç dij - dkkdij ÷ + b dkkdij
W
3
è
ø
• Analogous to Biot’s poroelasticity. (Poroviscosity?)
• Different from Newton’s law of viscosity
Alternative way to write the model
sij = 2heij ,
m
s m + dij = b dkk
W
change shape
change volume
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
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Inhomogeneous field
N = J + v /W
Net
flux
Diffusion
flux
Convection
flux
D
Ji  
 ,i
kT
1 æç ¶vi ¶v j ö÷
dij = ç
+
2 è ¶x j ¶xi ÷ø
Suo. Journal of Applied Mechanics 71, 77 (2004)
12
Boundary-value problem
4 partial differential equations
 ij , j  bi  0
N k,k = R
4 boundary conditions
s ij n j = ti - gk ni
m = -Wt j n j +Wgk
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
13
Length scale
L=
h DW
kT
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
14
Time scale
hL
G=
g
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
A cavity in a supercooled liquid
• A small object evolves by self-diffusion.
• A large object evolves by viscous flow.
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
16
Summary
1. A supercooled liquid is partially jammed. A drop in
temperature jams viscous flow, but does not retard selfdiffusion as much.
2. We regard viscous flow and self-diffusion as independent
processes, and formulate a “new” fluid mechanics.
3. A characteristic length exists. A small object evolves by
self-diffusion, and a large object evolves by viscous flow.
4. Other partially jammed systems: cells, gels, glasses,
batteries.
Li, Liu, Brassart, Suo. Journal of Applied Mechanics 81, 111007 (2014)
17