Pressure- and temperature- dependences of shape fluctuations in a microemulsion system

Hideki Seto (Yoshikawa Lab.)

Pressure- and temperature- dependences of shape fluctuations in a microemulsion system

Hideki SetoDepartment of Physics, Kyoto University, Japan

with collaborations of Michihiro Nagao ISSP, The University of Tokyo

Takayoshi Takeda FIAS, Hiroshima Univ.

Youhei Kawabata Tokyo Metropolitan Univ.

…and many other colleagues


Ternary microemulsion systemswater + oil + surfactant

Surfactant

Water Oil

Lamellar

Spherical Micelles Inverted Micelles

Irregular Bicontinuous

Hexagonal

Inverted CubicCubic

Cylindrical Micelles


R1

R2

mean curvature

€

H =12

1R1

+1R2

⎛

⎝ ⎜

⎞

⎠ ⎟

Gaussian curvature

€

K =1R1

1R2

Helfrich’s approachW. Helfrich, Z. Naturforsch. C28 (1973) 693

€

Ebend = κ (H −1Rs

)2 +κ K ⎡

⎣ ⎢

⎤

⎦ ⎥∫ dSBending energy

bending modulus

Spontaneous curvature


H > 0

H = 0

H < 0

K > 0 K = 0 K < 0


Phase transitions

spontaneous curvaturesbending moduli

SANS/SAXS and NSE studies

Phase transitions are observed with increasing temperature, pressure, ...

change with changing conditions


AOT + D2O + n-decane

AOT

decane D2O

φ

water-in-oil droplet

AOT molecule

spontaneous curvature > 0


T-(droplet volume fraction) phase diagram

0 0.1 0.2 0.3 0.4 0.5 0.6

T [˚C]

20

30

40

502 phase

1 phase

lamellae

binodal line

droplet

Cametti et al. Phys. Rev. Lett. 64 (1990) 1461.


Origin of temperature dependence

Rs > 0

Rs ~ 0

lamellar structure

Rs >> 0

w/o droplet

T


Pressure dependence

0 0.1 0.2 0.3 0.4 0.5 0.6

P [MPa]

0

10

20

30

40

percolation line

binodal line2 phase

1 phase droplet

Lamellae

Saïdi et al. J. Phys. D : Appl. Phys. 28 (1995) 2108.


SANS measurement

upper part

lower part

0

10

20

30

40

0 0.05 0.1 0.15

0.1 MPa10.230.544.557.0121.0

Q [Å-1]

I(Q) [cm

-1]

0

10

20

30

40

0 0.05 0.1 0.15

0.1 MPa10.230.044.558.0120.6

I(Q) [cm

-1]

Q [Å-1]

Nagao and Seto, Phys. Rev. E 59 (1999) 3169


Determination of P(Q) and S(Q)

P(Q):form factor of dropletpolydisperse droplet with Schultz size distribution

Kotlarchyk and Chen, J. Chem. Phys. 79 (1983) 2461.

(R0: mean radius of water core)

S(Q):inter-droplet structure factorhard core and adhesive potential

Liu, Chen, Huang, Phys. Rev. E 54 (1996) 1698

R ' R r

Ω

0

water AOT decane

L(Q)=1/(2Q2+1)

:surfactant concentration fluctuation

R0


Result of fittingφs = 0.230 / T = 33°C / P = 1bar

3500

3000

2500

2000

1500

1000

500

00.200.150.100.05

Q(Å-1)

I(Q)=P(Q)S(Q)+L(Q)

R=51.9(Å)=0.28Ω=-3kBT=0.0013Z=26.1R0=40.5 (Å)=10.6(Å)


Pressure dependence of Ω

-8

-7

-6

-5

-4

-3

-2

-1

0

-400 -200 0 200 400 600 800

T=20°CT=24°CT=29°CT=34°C

P-Ps(bar)

Ω(kBT)


Pressure-induced transition

pressure


Dynamical behaviorPressure-dependence Temperature-dependence

SAME? or DIFFERENT?

dilute droplet

Y. Kawabata, Ph. D thesis to Hiroshima Univ.

dense droplet

M. Nagao et al., JCP 115 (2001) 10036.

AOT

decane D2O

Temperature/Pressure

φ

dilutedroplet

densedroplet


Neutron Inelastic/Quasielastic Scattering

Low wavelength resolution

Low energy resolution

High resolution

Less intensity


Neutron Spin Echo

Larmor precession in a magnetic field

Wavelength resolution and engergy resolution are decoupled


Advantages of NSE

Highest energy resolution ~ neV

I(Q,t) is observed : better to investigate relaxation processes

BEST for SLOW DYNAMICS in SOFT-MATTER


Dense droplet

Q=0.09Q=0.07Q=0.06Q=0.05Q=0.03

Q=0.11

Fourier time(ns)0.3

0.4

0.5

0.6

0.70.80.9

1

0 5 10 15

Fourier time(ns)0.3

0.4

0.5

0.6

0.7

0.80.9

1

0 5 10 15

Q=0.04Q=0.06Q=0.07Q=0.09Q=0.10Q=0.12Q=0.14

Q=0.04Q=0.06Q=0.07Q=0.09Q=0.10Q=0.12Q=0.14

Fourier time(ns)0.3

0.4

0.5

0.6

0.7

0.80.9

1

0 5 10 15

25

40

0.1 60Pressure/MPa

I(Q, t)= Σ<p(-iQRj(0))p(iQRj(t))>N1


Model of membrane fluctuationZilman and Granek, Phys. Rev. Lett. 77 (1996) 4788)

h (kBT/κ)1/2 Lζ~ : roughness exponentζ = 1 (2 )D object= 3/2 (1 )D object

L (κ/kBT)1/2ζ Q-1/ζ~The Stokes-Einstein diffusion coefficient is,D(Q) (kBT/ηL) (kBT/η)(kBT/κ)1/2ζ Q1/ζ~ ~The relaxation rate is,

Thus they obtained the stretched exponential form of the relaxation function as,

where

I(Q, t )= exp[-(Γ(Q)t)β]

Γ(Q)= γαγκ (kBT)1/βκ1-(1/β)η-1Q2/β

β = 2 / (2+1/ζ) = 2/3(2 )D object= 3/4(1 )D object

γα = 0.024 (2 )D object = 0.0056 (1 )D object

γκ = 1 - 3 ln( / l(t)) kBT / (4pκ)

~Γ(Q) D(Q)Q2 (kBT/η)(kBT/κ)1/2ζ Q2+(1/ζ)~

Q // z

undulation amplitude: h = 1/Q

lateral length: L


Bending modulus

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.5 1 1.5 2 2.5 3

T=25°C / P=0.1MPaT=40°C / P=0.1MPaT=25°C / P=60MPa

Q 3 (x10 -3Å-1)

Γ(1/ns)

κhigh-T

κambient-T,P

κhigh-P

0.4kBT

1.4kBT

2.6kBT

Γ(Q)= 0.024(kBT)2/3 κ 1/3 η -1 Q3


Dilute droplettemperature / pressure

AOT / D2O / d-decane (film contrast)

s=0.37 (AOT volume fraction)

=0.1 (droplet volume fraction)


Measured pointsAOT / D2O / d-decane (film contrast)

s=0.37 (AOT volume fraction)

=0.1 (droplet volume fraction)

1 0

2 5

4 3

5 55 9

6 5

0 .1 2 0 4 0 6 0P /M P a

T /°C


Result of SANS

1

10

100

I(Q

) [c

m-1

]

0.012 3 4 5 6 7 8 9

0.12

Q [Å -1

]

T=298.15K

P=22 MPa

T=298.15K P=0.1MPa

T=329.15K

P=0.1MPa€

F(Q) = Fmono (Q,R)exp[−(R − R0 )2

2σ 2−∞

∞

∫ ]dR

Fmono(Q) =sin(QR)

(QR)

⎡

⎣ ⎢ ⎤

⎦ ⎥

2

p =σR0

R0 ~ 32Å → 28Å

T=25 ˚C → 65˚C

P=0.1 MPa → 60 MPa

R0 ~ 32 Å → 30Å

p ~ 0.16 → 0.18

p ~ 0.16


NSE profiles

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15

I(Q=0.04)I(Q=0.05)I(Q=0.06)I(Q=0.08)I(Q=0.09)I(Q=0.10)I(Q=0.12)I(Q=0.14)

t [ns]

€

I(Q,t)/I (Q,0) =exp[−DeffQ2t]

Room temperature/pressure

1.0

0.8

0.6

0.4

0.2

I(Q,t)/I(Q,0)

1412108642 t [ns]

'0.04' '0.05' '0.06' '0.08' '0.09' '0.10' '0.12' '0.14'

T=43˚C/ P=0.1MPa

T

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

I(Q,t)/I(Q,0)

1412108642t [ns]

'Q=0.05' 'Q=0.06' 'Q=0.07' 'Q=0.08' 'Q=0.09' 'Q=0.095' 'Q=0.10' 'Q=0.11' 'Q=0.12' 'Q=0.13'

RT/ P=20MPa

P


Milner and Safran modelHuang et al. PRL 59 (1987) 2600.Farago et al. PRL 65 (1990) 3348.

€

R(θ ,φ,t)=R0{1+ anm(t)Ynmnm∑ (θ ,φ)}

Expansion of the shape fluctuation into spherical harmonics

€

f0(QR0)=[ j0(QR0)]2

f2(QR0)=5[4 j2(QR0)−(QR0) j3(QR0)]2

up to n=2 mode gives

€

I(Q,t)/I (Q,0)=exp[−DeffQ2t]

Deff=Dtr+5λ2f2(QR0) a2

2

Q2[4πf0(QR0)+5f2(QR0) a22

]

where

damping frequency of the 2nd mode deformation

mean-square displacement of the 2nd mode deformation

n=0 mode n=2 mode

translational diffusionshape deformation


Effective diffusion constant

12

10

8

6

4

2

0.140.120.100.080.060.04Q [Å-1]

T= 19˚CT= 25˚CT= 35˚CT= 49˚CT= 55˚C

P= 60MPa

P= 21MPaP= 40MPa

Def

f [1

0-7 c

m2 /

s]

temperature

pressure

€

Deff=Dtr+5λ2f2(QR0) a2

2

Q2[4πf0(QR0)+5f2(QR0) a22 ]


Expansion of the theory

κ

λ2= κηR0

3 4R0Rs

−3κκ−3kBT4πκ f φ( )⎡

⎣⎢⎤⎦⎥

1Z2( )

p2= kBT4π 62κ+( )−8κR0

Rs+3kBT

2π f φ( )⎡⎣⎢

⎤⎦⎥

EXPERIMENTALY OBTAINED PARAMETERS

KNOWN PARAMETERS

Seki and Komura Physica A 219 (1995) 253 ηη

€

Z(2)=23

′ η η

+32

24

€

κ =16

kBT

8πp2+λ2R0

3ηZ(2)⎛

⎝ ⎜

⎞

⎠ ⎟

Y. Kawabata, Ph. D thesis

€

R0 ≈ 32Å

€

p2 = σR0

⎛ ⎝ ⎜ ⎞

⎠ ⎟2 ⎛

⎝ ⎜

⎞

⎠ ⎟≈ 0.16

From SANS experiments


Pressure- and temperature-dependence of κ and <|a2|2>

(A) : Temperature dependence of κ (B): Pressure dependence of κ

0.55

0.50

0.45

0.40

0.35

0.30

0.25

κ [kB]T

330320310300290 [ ]Temperature K

3.0

2.5

2.0

1.5

1.0

0.5

< |a2|2>

κ

< |α2|2>

( )A3.0

2.5

2.0

1.5

1.0

0.5

< |a2| 2>

6004002000Pressure [Kg/cm2]

0.55

0.50

0.45

0.40

0.35

0.30

0.25

κ [kB]T

( )B κ < |a2|

2>


Introducing reduced pressure / temperature

3.0

2.5

2.0

1.5

1.0

0.5

< |a2|2

>

-1.0 -0.5 0.0 0.5T, P^ ^

(B)< |a2|2>(pressure)

< |a2|2>(temperature)

TB , PB : binodal point

T0 , P0 : ambient temperature/pressure

€

ˆ T =T −TB

TB −T0

ˆ P =P−PBPB −P0

binodal pointambient temperature/pressure

0.50

0.45

0.40

0.35

0.30

0.25

κ [kB]T

-1.0 -0.5 0.0 0.5, T P

κ ( )pressure

κ ( )temperature

( )A

^ ^


Schematic picture

Temperature Pressure


Pressure- and temperature dependences of head area

64

62

60

58

56

54

52

-0.8 -0.4 0.0 0.4

T, P^ ^

a H[Å

2 ] a H=area per molecule

number of surfactants per droplet

number of droplets =Whole volume of droplets

volume of a droplet

number of surfactants per droplet number of droplets

number of surfactants=

temperature

pressure


SummaryPressure- and temperature-dependences of the structure and the dynamics of AOT/D2O/decane were investigated.

bending modulus for Gaussian curvature κspontaneous curvature Rs

κ increase decrease

microscopic tail-tail interaction counter-ion dissociation

pressure temperature

structuredense droplet lamellar/bicontinuousdilute droplet 2-phase droplet

Pressure- and temperature- dependences of shape fluctuations in a microemulsion system

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