Post on 12-Apr-2022
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)
AMPERE, Zakopane, 2018 2
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
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
Velocity autocorrelation function
Molecular self-diffusion in the magnetic fieef gradient
AMPERE, Zakopane, 2018 4
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
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Self-diffusion measurement by PGSE
AMPERE, Zakopane, 2018 6
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
AMPERE, Zakopane, 2018
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
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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
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, ,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
AMPERE, Zakopane, 2018
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
Velocity autocorrelation function
d =6 ms t =
PGSE in Gaussian approximation
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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
AMPERE, Zakopane, 2018 12
, ,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
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2.5msd
, , ,i
i
iq zFT E q E q e dq P zt t t
Probability distribution
AMPERE, Zakopane, 2018 14
, ,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
AMPERE, Zakopane, 2018
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
AMPERE, Zakopane, 2018 16
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
AMPERE, Zakopane, 2018
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.
Velocity autocorrelation function
17
AMPERE, Zakopane, 2018
Velocity autocorrelation function
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
AMPERE, Zakopane, 2018
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
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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
AMPERE, Zakopane, 2018
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
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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
𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦
𝜋/2𝑥
𝜋𝑦 𝜋𝑦 𝜋𝑦
MGSE measurement of water
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2t
𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦 𝜋𝑦
𝜋/2𝑥
𝜋𝑦 𝜋𝑦 𝜋𝑦100MHz NMRGt = fixedwater
40
300
600 echoes
MGSE of diffusion liquids
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tolueneethanol
2
log 1d E
dt T
water
MGSE of diffusion water
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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
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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
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Hydrodynamic fluctuation and Long time tail of velocity autocorrelation
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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
AMPERE, Zakopane, 2018
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
AMPERE, Zakopane, 2018 29
3 / 2D t t D
ethanol
Experimental confirmation of hydrodynamic fluctuation in liquids
by NMR gradient spin echo??
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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
AMPERE, Zakopane, 2018
Aleš Mohorič
Igor Serša
Franci Bajd
Ulrich Scheler,
Bernd Fritzinger
Carlos Mattea
Siegfried Stapf
31
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