CMS summer school on ‘Microstructure: evolution and dynamics’
Technion, Aug. 25-29, 2013
Statics and dynamics of phase transitions in electric fields
Jennifer Galanis, Sela Samin, and Yoav Tsori
Ben-Gurion University of the Negev
• E-field:
(De)2
liquid 2 E0
liquid 1
2
21
2
21
2
02
||
||
216
9
ee
ee
e
ER
RR
RR
R┴
R||
O'Konski & Thacher '53
Allan & Mason '62
e2 e1
problem: sometimes drop is oblate 55
3213212
M
MRDDR G. I. Taylor '66
D, R, M - ratios of e, r, n of liquid 2 to liquid 1
>0 : prolate <0 : oblate
Leaky dielectric model - conductivity matters !
• interfacial tension: area → perfect sphere
energy penalty if dielectric interfaces perp. to E
r32d2
1Ee → needle-like drops
Dee2-e1
Dielectric drops in electric fields Introduction
e2
e1
E0
E0 destabilizes film
h0
xhhxh 0
/cos teqxh
22*23
0 23
1qqq
h
0
2/1** ~/2
Eq
e
Dfastest growing wavelength:
h
q ~
1/
~
1
2
3
4 5 6
7
1 - e1/e2=2
2 - e1/e2=5
3 - e1/e2=10
4 - e1/e2=2 S«1
5 - e1/e2=5 S«1
6 - e1/e2=10 S«1
7 - S=
S. Herminghaus, PRL 99' W. B. Russel, 2002
x
linear stability analysis:
T. Russell, U. Steiner, Nature (2000)
growth rate:
Normal field instability Introduction
Wetting angle decreases: q*<q
d
V
eqq
2coscos
2*
modified Young-Duprè eq.:
Lippmann, B. Berge
Electrowetting
R. S
ham
ai et al., S
oft
Matter
2008
VariOptic Ltd. Plastic Logic Co.
Variable focal length lenses Electronic paper
Introduction
Orienting BCPs
Russell’s group, Science ’96
No field Annealed in electric field
electrode
E
electrode
"good" "bad"
E eB
eA
eA
eA eA eB eB eB L E
Why lamellae are oriented ?
2
0
2
esEU
BAee Energy penalty if dielectric interfaces perp. to E:
Introduction
Can an electric field create an interface ?
Intermolecular potential Classical P-V curve
Can an electric field change the classical van der Waals picture ?
r
U(r)
V
P
...50
)5(30
)3(0
)1( EEEp eee● Nonlinear polarizability:
● Dependence of e on r :
● Resonance ww0: 220
)1( 1~
ww
e.g. w0~6·1015 sec-1 (=300 nm)
(3) : Laser self-focusing, nonlinear optics (femtosecond lasers)
r
e
e
e
eeee
0
20
2
3
22
Tkg
B
k
Onsager 1936
Clausius-Mossotti
(1) : linear susceptibility ~ atomic volume ~ (Å)3
Tc is renormalized by:
rUes32d
2
1Ere
Constitutive relation er
...''2
1 2D ccc rrerreee
mK102
'' 20 D
B
ck
EvT
e
Theory: Landau & Lifshitz, Electrodynamics of Continuous Media (1st Ed):
A fit to e(r):
0
T
r1/V
phase-diagram
rc,Tc)
Liquid-vapor coexistence in uniform fields
Uniform field: effect is local
DTc
Electrostatic energy:
Debye & Kleboth (1965)
e
r
Low e High e De DT [mK] E [V/m] Ref
Iso-octane
2.0
Nitrobenzene
34.2
32.2 -15 4.5 Debye 1965
Orzechowski
1999
Cyclohexane
2.0
Aniline
7.8
5.8 -80 O.3 dc Beaglehole
1981
Cyclohexane
2.0
Aniline
7.8
5.8 0 1 Early 1991
n-hexane
2
Nitroethane
19.7
17.7 -20 1 Wirtz &
Fuller 1993
Experiments: Tc Tc-15 mK
● Small effect * (removed from L&L)
new type of phase-transitions
Nonuniform field E(r):
● Disagreement between
theory & experiment
● Dimensionality & sign problems
* Exception: S. Reich & J. M. Gordon, J. Pol. Sci.: Pol. Phys. 17, 371 (1979). Polymers: v0 → Nv0
mixture of liquids
1 & 2
Surprise - an interface appears
liquid 1
liquid 2
V T/Tc
"electro-
prewetting" prewetting
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9
0.95
1
wetting
electrowetting
f : fraction of liquid 1
binary mixtures / liquid-gas coexistence
veE2«kBT - no chance for phase separation ?
Demixing in field gradients
Electric field: a truly long range force:
Field E changes density r
Density r affects E far away Laplace's equation
e1 e2
E Displacement of the interface: energy volume
·area not good !
"Surface tension"
𝛻 휀 𝜌 𝛻𝜓 = 0
02
1 2
r
e
r
r
bFF1)
22
1re bFF
bulk free energy
Equilibrium:
Free-energy:
r electrostatic potential
(r)=?
r(r)=?
Problem formulation
0 re
F2)
03d
1rr r
Vr3)
1
2
Q3
r 0
Fb
coupled nonlinear equations
we are here
0
T
r r0
E=- electric field
Phase transitions in field gradients
Fb
23 1log1log rrr abTkF Bb
r : density
a, b : interaction, hard core volume parameters
e.g. van der Waals:
: de Broglie wavelength
r
VE
q q
r
V
R1
R2
1) Charged colloid
2) Charged wire/
Concentric cylinders
3) Wedge
r
r
R2
R1
s: surface charge density
s: surface charge density
R1
fixed charges or potentials "open" or "closed" systems
s
V: potential difference
q: opening angle
Field gradients: three "canonical" systems
2
2
1
r
RE
re
s
r
RE 1
re
s
E || r
E || r
E r
Fb'(r) er
D r
Fb 2
2
1E
1
2
2
1REe D
r
2
2
2
1REe D
R2 r
r(r)
R1
0.4
0.8
1.2
1.6
V<Vc
V>Vc
1
2
2
1REe D large voltage
0.95
r
E=0
non-uniform E uniform E (Landau)
"electro prewetting"
Field-induced phase transition graphical solution
small voltage
0.8 1 1.2
0.9
0.95
1
r/rc
T/Tc
DT
1st order, R=R1
binodal
Stability diagram
2
1
0 1
2
DD
R
V
k
vT
cB qrr
e
displacement of coex. temp.:
YT, F. Tournilhac & L. Leibler, Nature (2004)
S. Samin & YT, J. Chem. Phys. (2009)
2nd order, finite R
Charged wire
V2>Vc: V1<Vc:
R
r/R
1.5 2 1 0.8
1
1.2
r/r
c gas liquid
V1<Vc
V2>V1 : larger potential
V3>V2 : even larger potential
Liquid-vapor coexistence
rr d,,, rr TfTfFesvdw
23 1log1log, rrrrr abTkTfBvdw
Free energy:
van der Waals energy:
Nucleation of liquid droplets
Laplace formula:
Rpp
outin
Classical case: E=0
jiijij DEEf
EfP
r
r
rr ,
,
Pressure tensor:
● Pressure is uniform
inside/outside
● pin > pout
Prr : Stress in radial direction
Pqq : Stress in azimuthal direction
● Pressure is nonuniform
● pin < pout
S. S
am
in &
YT
, J. P
hys. C
hem
. B
(2010)
Funny bubbles & drops
1 1.5 2
0.8
1
1.2
r/r
c
r/R
Density
r/R
1.5 2 1 0.93
0.95
0.97
Prr
Pqq
Pressure tensor
P/P
c
V1
V2
V3
V1 <V2 <V3
-40 -20 20
0.8
0.9
1
1.1
1.2
r
Gibbs dividing
surface
r/r
c
2
1
sharpsmoothsmoothsharp
R
R
R
R
drdr rrrr
area
sharpsmooth
1
rr
RPc
3
r D
0.15 0.25 0.35 0.45
0
0.05
0.1
0.15
Dr
rsmooth(r)
Gibbs dividing surface occurs exactly at
the same R as in the sharp interface limit
Equal shaded areas: Surface tension:
The same as in the absence of field
(but Dr is different)
Dr
rsharp(r)
Density profiles Surface tension
Surface tension
1.2
1.15
1.1
1.05
1
+
+ -
- V=10 V=11 V=12 V=13 V=14 V=15 V=16 V=17 V=18 V=19 V=20 V=21 V=22
T=0.999Tc
r=1.08rc
R=2.5 m
● Nonspherical bubbles
● What is the contact
angle at the wall? a
liquid
gas
Liquid-vapour coexistence
Potential difference
between + and - :
volume & surface tension
depend on E !
0
T
r
r/rc
S. Samin & YT, J. Phys. Chem. B (2010)
Nucleation of gas bubbles
Switching to liquid mixtures
a b
d c
V=0, T=Tcoex+1°C V=0, T=Tcoex -0.3°C
V=200V, T=Tcoex+0.3°C V=300V, T=Tcoex+0.5°C
50 m
glass
slide
Liquids:
Silicon oil & Paraffin
(PMPS & Squalane)
Experiments in liquid mixtures Leibler Lab, ESPCI
ITO electrode
100 V
1 kHz
microscope
Z
X
Y
Cover slip
Log T-Tcoex [°C]
Lo
g V
* [v
olt]
V [Volt]
R [
m
]
2/1
2/1
0
1
* 2coexc
B TTv
kRV
D ff
eq
Width of wetting layer Critical voltage
0 100 200 300
0
10
20
30
40
DT=1 °C
DT=0.5 °C
DT=0.2 °C
DT=0.1 °C
a
2/1
coex
TTaslope:
YT, F. Tournilhac, L. Leibler, Nature (2004)
V
R
V
TTk
vRR
ccoexB
D
2/12
1
0
1
2 qff
e
-2 -1 0
3
4
5
0.5 theo. slope
best fit to exp.:
0.7±0.15
R width of wetting layer
Liquids: silicon oil & paraffin
(PMPS & Squalane)
Spinodal decomposition ? Nucleation and growth ?
ITO electrode
100 V
1 kHz
microscope
Z
X
Y
Need to develop a new model !
"razor-blade" electrodes
NO !
Rotational symmetry: broken
Model for phase separation :
● Shape, size & velocity of droplets, scaling laws, importance of viscosity ?
● AC electric fields - Screening length depends on the frequency w
Translational symmetry: broken
Dynamics
Birth of an interface !
YT & F. Tournilhac, Leibler Lab.
x
y
top glass
electrode
10 m
Field on (t=5 sec) Field off (t=5 sec)
reversible transition
60 m
V
Cascade of voltages:
Vc(1): demixing
Vc(2)>Vc
(1): 1st interf. instability
Vc(3), Vc
(4) , , , ?
Not Rayleigh instability !
Interfacial instabilities
Droplet formation
Fingering instability
competition between surface
tension and E-field
• Purely dielectric field gradients depend on curvature
• Ions large screening - curvature not important
Demixing in polar liquids G
. Hefte
r. Pure
Appl. C
hem
. (2005)
DG for F- ion in water+solvent mixtures DG for H+ ion in water+solvent mixtures
MeOH
EtOH
EG
DMF
FA
MeOH
AC
MeCN
DG= Gibbs transfer energy for moving an ion from solvent 1 to solvent 2 • Preferential solvation:
Onuki PRL 2009. PRE 2010 Ben-Jacob et al., Curr. Opin. Colloid. Interf. Sci. 2011
M. E. Leunissen et al. PNAS 2007, Nature 2005, PCCP 2007 Zwanikken & van Roij PRL 2007
dielectrophoretic force
E~V/R
E~V/D
authoritative book
Pers
son. J
. Chem
. Soc. F
ara
day T
rans. (1
994)
Preferential solvation of ions in solvents
Du+ : how much a (+) ion prefers liquid 1 over liquid 2
n± : number density of positive/negative ions DG+Du+n+f DG-Du-n-f
DG: Gibbs transfer energy for moving an ions from solvent 1 to solvent 2
D
D
D
Tk
eVvn
Tk
u
T
T
BcBc
exp00
ffe
e
dielectrophoretic electrophoretic
small V
ffef DD nununnnnTkennFFBb
1ln1ln2
1 2
Free-energy:
YT & L. Leibler, PNAS 2007
Change to coexistence
temperature DT : demixing greatly enhanced
in polar liquids
If potential is small E is small not strong enough to induce electro-prewetting
Water-lutidine mixture (D=10nm): V=150 mV DT>100 K
large V
x
f(x)
Demixing in polar liquids
Very nonlinear problem (Debye-Hückel not good)
1. Steric stabilization
Graft/adsorbed polymers, surfactants
2. Electrostatic stabilization
Screened Coulomb potential (Yukawa):
DDeD
eDU
e/
2
TkenBoD
e /22
Debye screening length D=10nm (1 mM NaCl in water)
e e
D With salt
“Application”: colloidal stabilization
Two standard ways to stabilize colloids against van der Waals attraction:
U(D)
D
more salt DLVO theory
Van der Waals attraction
Electrostatic repulsion
How else can we stabilize colloids ?
Stabilization by addition of salt
In the absence of salt: colloids stick together, sediment
With “regular” salt: colloids stick together, sediment
Put colloids in a binary mixture
Add antagonistic salt: repulsion, colloids are stabilized
Requisites
Antagonistic salt: anion likes water, cation likes lutidine (or vice versa)
Colloids chemically prefer one of the solvents
“useful” case: neutral colloids (uncharged)
D
What is the potential U(D) ?
Osmotic pressure:
Effective potential between colloids:
Π = 𝑇𝑛𝑚 − 𝑇𝜔mix 𝜙𝑚 − 𝑃𝑏
𝑈 𝐷 = − Π 𝐷′ 𝑑𝐷′𝐷
∞
How we calculate the total effective colloid potential ?
Calculate profiles: 𝛿𝑓
𝛿𝜙= 0,
𝛿𝑓
𝛿𝑛±= 0,
𝛿𝑓
𝛿𝜓= 0
Add van der Waals 𝑈𝑣𝑑𝑊 = −𝐴
12𝐷2
s
dfdfffF rresionmixFree energy:
2
mix
3
2
111log1log ffffffff CTkfa
B
s
f rf
D
2
es
2
1fe f
mixture
ions
electrostatic
interfacial
𝑓ion = 𝑘𝐵𝑇 𝑛+ log 𝑎3𝑛+ − 1 + 𝑛− log 𝑎3𝑛− − 1 -𝑘𝐵𝑇 ∆𝑢
+𝑛+ + ∆𝑢−𝑛− 𝜙
Stabilization by addition of salt
Reminder of DLVO behavior
In our theory:
5 10 15 20 25 30 -10
-5
0
5
10
U/k
BT
D [nm] D [nm]
10 20 -10
-8
-4
0
4
8
12
U/k
BT
T-Tc=1.6K
T-Tc=3.2K
T-Tc=6.4K
T-Tc=9.6K
T-Tc=12.8K
nc=0.004M
nc=0.02M
nc=0.1M
Varying temperature Varying salt conc.
U(D)
D
more salt
0.5 0.55 0.6 0.65
-10
-5
0
5
10
15
20
3
3
5
5
7
10
3040
50
5
10
15
20
25
30
35
40
45
50
0 10 20 0
10
20
f0=0.46
f0=0.5
f0=0.65
Um
ax/k
BT
T-Tc [Ko] f0
T-T
c [
Ko]
D Dmax
Umax
Barrier height Umax/kBT Barrier height Umax/kBT
f0=0.46
f0=0.5
f0=0.65
weak dependence on T, all T
weak dependence far from Tc, strong close to Tc
strong dependence, all T
Stability regions
below the binodal
Maxwell's equations + currents: dispersion relation ww(k)
Extremely rich physics !
● Transverse waves:
"good" conductor (s0/w»e) : screening length =1/Im(k)~(s0w)-½
● Longitudinal waves:
"poor" conductor (s0/w«e) : screening length ~ independent of w
w>wp: field propagates
: plasma frequency
w<wp: field decays
Challenge - coupling between waves and phase separation
General picture:
● High frequency: field gradients originate from geometry (dielectric)
● Static fields: Poisson-Boltzmann, screening (conducting)
● Finite frequency: unknown
w wp
Re(k)
Im(k)
Time-varying fields: electromagnetic waves
tieEE w rk0
ew
m
nep
22
2222 kvtp ww
k
ffef DD nununnnnTkennFFBm
lnln2
1 2
Theoretical framework
T
Ennq
fP
t
fL
t
r
er
f
fr
fe
f
f
f
2
2
1
0
0
E
vvvv
v
v
Dynamics of liquid phase separation:
BAkuCkT
DCD
dt
Cd
BAkuBkT
DBD
dt
Bd
BAkuAkT
DAD
dt
Ad
C
C
C
B
B
B
A
A
A
2
2
2
][
][
][
Chemical reactions:
rrr ff 121 uuuA
Electromagnetic waves and currents:
Evj
j
qnnDn
t
n
0
0
BjjD
H
DB
E
t
nnqt
Temperature:
jii
j
j
i
pr
u
r
uqqTT
t
TC
,
2
2
1kr EjEjv
Free energy:
Outlook – phase transitions in field gradients
● Demixing in electric field gradients: liquid-liquid and liquid-vapor coexistence
● Fascinating dynamics & interfacial instabilities
● Microfluidics of mixtures: demixing should always occur
● Stability of colloidal suspensions: non-DLVO behaviour
Field gradients → layers
Charged moving parts
→ huge reduction of friction
MEMS, NEMS:
Friction becomes dominant as size shrinks
a factor of 1/2 = 1500 for water/glycerol
"Inverse problem" of microfluidics - how to demix ?
separate soluble organic solvents from polluted water
Lab-on-a-chip: separate nicotine or alcohol from water
● Demixing and additional hydrodynamic flow
● Electro-lubrication
Preliminary dynamics
New type of interfacial instabilities: E-field stabilizes interface, surface tension de-stabilizes
Electro-prewetting around a charged cylinder
Model B dynamics:
J. Galanis & YT
𝜕𝜙
𝜕𝑡= 𝐿𝛻𝜇 𝜇 =
𝛿𝐹 𝜙,𝜓
𝛿𝜙
𝛻 휀 𝜙 𝛻𝜓 = 0
