Structure of porous media and hydrodynamic fluctuations in ...

Structure of porous media and hydrodynamic fluctuationsin liquids observed by NMR gradient spin echo methods

Janez Stepišnik

University of Ljubljana, FMF, and J. Stefan Institute, Ljubljana, Slovenia

AMPERE, Zakopane, 2018 1

2 2 2 2

1 1 1

0 0

2

j i j

n n n

n j i i

j i j

X X X i j

tX X X X n X X D t

t

Brownian motion (Robert Brown in 1827)


Albert Einstein1879-1955

Nobel Price 1921

Jean Baptiste Perrin1870-1942

Nobel Price 1926

n

j

jnnXXXXXX

1

321....

1 0

2

0 0

( ) ' '

0 0 ( ) ' . " " '

tn

n

j

t t

n

X t v t t v t dt

X v t X t v t v t dt dt

Self-diffusion coefficient

Velocity autocorrelation function

http://en.wikipedia.org/wiki/File:Einstein_1921_portrait2.jpg

Molecular self-diffusion and velocity autocorrelation

AMPERE, Zakopane, 2018

1

2

1 2 2 1

0 0

( ) 2 ( ) ( )

tt

x xx t v t v t dt dt

3

Brownian motion is governed by collisions with the particles in the medium, while the molecular self-diffusion is determined by molecular interactions,but the particles move randomly in both cases.

Einstein´s definition of selfdiffusion coefficient:

Green-Kubo relation:

2

0

2

2

2

1lim ( ) ( ) (0)

2

1( ) (0) ( )

2

xx x xt

x x

dD x t v t v dt

dt

dv t v x t

dt


Molecular self-diffusion in the magnetic fieef gradient


History:

E. L. Hahn, Phys. Rev., 80, 580 (1950)H. Y. Carr and E. M. Purcell, Phys. Rev., 94 , 630-38 ,1954D. W. McCall, D. C. Douglass, and E. W. Anderson, Ber. Bunsenges. Physik. Chem. 67, 336 (1963). H. C. Torrey, Phys. Rev., 104, 563-565, 1956 Stejkal, E. O and Tanner, J. E., J. Chem Phys., 42, 288 (1965)

G B r

Magnetic field gradient

Pulsed Gradient Spin Echo (PGSE)

t

Stationary spins

0

.

1i

i

i

i t f t dt

E e

t

t

G r

RF

G

d d

f t


Self-diffusion measurement by PGSE


6

0

.

1i

i

i

i t t f t dt

E e

t

t

G r

t

Moving spins

d d

G f t

Spin motion encoded in the spin phase


0i

i

i

i t t f t dt

E e

t

t

G r

0

' ' 't

t t f t dt q G

Spin phase discord

0

.i

i

i t t dt

E e

t

t

q v

Velocity is a random variable

Velocity of spin translation

q(t) Geff

G

7

t

f t

0

i

i

i t t dt

e

t

q v

Integration by parts


i

0

.

i

i t t dt

E e

t

t

q v

spin echo

/2

dd

q(t)

0

i

i

i zi

iq zE e

q G z v t dt

t

tt

d t

Propagator method

. ' . ' . ' ....

t

i i i

o o o

i

i t t dt t t t t dt dt

e

t t

q v q v v q

Cumulant expansion:Gaussian phase approximation

t

,i

i

iq zP z e dVt

PGSE with the propagator method


, ,i

i

iq zE q P z e dV

q G

t t

d

spin echo

/2

dd

2

i

i

q De

t

J. Karger and W. Heink. The propagator representation of molecular transport in microporous crystallites , J. Magn. Reson. 51, 1-7 (1983)

t

2

1 4,

4

ii

i

z

DP z e

D

tt

t

.

00 0

. . ' '. ...

t

i i i

i

i t . t dt t t t t dt dt

e

t t

q v q v v q


Cumulant expansion in the Gaussian phase approximation

Gaussian phase approximation

i

iiie

tt

ph

ase

gra

tin

g

i

0

.

i

i t t dt

E e

t

t

q v

1 1z

G q d

10


d =6 ms t =

PGSE in Gaussian approximation


2

i

0

. ' . ' ....

t

zi zi zi

o o o

i i

i t t dt iq v t dt q v t v t dt dt

E e e

t t t

t

q v

spin echo

/2

dd

t

Diffusive diffraction

P. T. Callaghan, A. Coy, D. MacGowan, K. J. Packer and F. O. Zelaya, Diffraction-like effects in NMR diffusion studies of fluids in porous solids, Nature , 351, 467-9 (1991)

q

2 21....

2i i

i

iq z q z

e

t

0i

z t

2....

i i

i

iq z q De

t t

PGSE of diffusion in the sieve made of polymer membrane


, ,i

i

iq zE q P z e dV q Gt t d

membrane2.5msd

spin echo

/2

dd

t

q-Fourier transform probability distribution


2.5msd

, , ,i

i

iq zFT E q E q e dq P zt t t

Probability distribution


, ,ii

i

iq zE q e dq P zt t

2 2 2

2 2 21 2 32 2 2

1 2 3...

z z z

z z z

a e a e a et t t

z


Distribution of pores in polyamid membrane

2

4

14

[ , ] c i

Dz

k rFT E q e

d t

d

d

d

t

2z

d

2ri

2.5msd

cd t

z

2z const

Pores in polymer membrane


membrane

J. Stepišnik, B. Fritzinger, U. Scheler and A. Mohorič, Self-diffusion in nanopores studied by the NMR pulse gradient spin echo , EuroPhysics Letters, 98 (2012) 57009

00 0

1' ...

2i i i

i

i t . t dt t . t t' . t' dt dtc

e

tt t

q v q v v q


Gaussian phase approximation of gradient spin echo

Gaussian phase approximation

i

iiie

tt

i

0

.

i

i t t dt

E e

t

t

q v

0

0

0

,

i i i c

i tt e dt

i tt e dt

t

t

D v v

q q

Velocity autocorrelation spectrum

Spectrum of spin dephasing

0

*,..,

1tt

t d

iiqDq

J.Stepišnik, Analysis of NMR self-diffusion measurements by density matrix calculation, Physica 104B (1981) 350- 361.


17



Interactions with boundaries set the long tail of VAF

VAF of gas, liquid

time

Ve

loc

ity

au

toc

orr

ela

tio

n tc~10-12-10-9 s

)(20 tDvtv d

VAF of restricted diffusion

Ve

loc

ity

au

toc

orr

ela

tio

n

time

Molecular velocity autocorrelation contains details of molecular interactions.

18

NMR

1 kHz

Spec

tru

mo

fve

loci

tyau

toco

rrel

atio

n

frequency

1 kHz

NMR

Spec

tru

mo

fve

loci

tyau

toco

rrel

atio

n

frequency

0

0 dtevtvDti

DdtvtvD

0

00

2

0 0

1,

i i

i

i t . t dt D d

E N e

t

t

t

q v q

Modulated Gradient Spin Echo


2

2

2 2

8i m

m

i

G D n

e

t

2 2 2

2

8, ....

m

m

nn

t t d

q G

q(t)

19

CPMG + fixed gradient

Use of MGSE method on different devices


2t

𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦

𝜋/2𝑥

𝜋𝑦 𝜋𝑦 𝜋𝑦 100MHz NMRGt = fixedwater

Gradient generated by the susceptibility differences in a porous medium - cement

Without applied gradient

2

2 2

4

8i m

i

G D t

E t e

t

hydrogel

MGSE measurement of restricted diffusion by 100 MHz NMR


D

r

D

rb

D

r

bDbDp

k k

k

kp

4

2

1

4

1

4

1112

1

2

1

22

1

2

1

122

2

rest

0.046)2(

2;08.2;:sphere

11D

t

t

t

t

t

t

0.0 0.5 1.0 1.5 2.0 2.5 3.01.0

1.5

2.0

2.5

3.0

3.5

kHz

Dx10

9m

2s

1

hydrogel 1

Dp=1.06×10-9 m2s-1

2r =2.57×10-6 m

0.0 0.5 1.0 1.5 2.0 2.5 3.02.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

kHz

Dx10

9m

2s

1

hydrogel 2

Dp= 2.38×10-9 m2s-1

2r = 2.42×10-6m

Theory

Dp

21

2t


𝜋/2𝑥

𝜋𝑦 𝜋𝑦 𝜋𝑦

MGSE measurement of water


2t


𝜋/2𝑥

𝜋𝑦 𝜋𝑦 𝜋𝑦100MHz NMRGt = fixedwater

40

300

600 echoes

MGSE of diffusion liquids


tolueneethanol

2

log 1d E

dt T

water

MGSE of diffusion water


2

log 1d E

dt T

2

2

logd E

dt 2

i

tk D t

T

i

i

Log E t Log a e

2

2 2

2

22

...2

variance; meani

t kA k D t D t

T

D D D D

Distribution of diffusion coefficients


2 2

2

log 1, ...

d Ek D k D t t

dt T

2

2 2

2

log, ...

d Ek D t

dt

2

2 2

2

...2

t kLog E t A k D t D t

T

MGSE of water by NMR MOUSE


Hydrodynamic fluctuation and Long time tail of velocity autocorrelation


1. V. Vladimirsky, J. Terletzky, J. Exp. Theoret. Phys. 15,258 (1945)2. L. Landau, E. Lifshitz, Fluid Mechanics (Pergamon Press,Oxford, 1987)3. M.S. Giterman, M.E. Gertsenshtein, J. Exp. Theoret.Phys. 23, 723 (1966)4. B. Alder, T. Wainwright, Phys. Rev. Lett. 18, 988 (1967)5. B. Alder, T. Wainwright, Phys. Rev. A 1, 18 (1970)

3 / 2D t t

Ve

loc

ity

au

toc

orr

ela

tio

n

time

D

Velocity autocorrelation and hydrodynamic fluctuation


D

D2

28

0 0

'

t

i i it . t t' . t' dt dt

t

t t q v v q

Long time tail of the velocity autocorrelation function


3 / 2D t t D

ethanol

Experimental confirmation of hydrodynamic fluctuation in liquids

by NMR gradient spin echo??


J. Stepišnik, C. Mattea, S. Stapf, A. Mohorič, Molecular velocity auto-correlation of simple liquids observed by NMR MGSE method, arXiv:1010.1175v3 [cond-mat.soft] JEPB

Molecular simulation

NMR MGSE

Acknowledgement


Aleš Mohorič

Igor Serša

Franci Bajd

Ulrich Scheler,

Bernd Fritzinger

Carlos Mattea

Siegfried Stapf

31

DZIĘKUJĘ BARDZO ZA UWAGĘ !

Structure of porous media and hydrodynamic fluctuations in ...

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