Pore Geometry from the Internal Magnetic Fields

Experiment and Simulations of NMR Echo Trains

for «Porous Solid / Liquid» System

by Dr Alexander SagidullinKTH – Royal Institute of Technology, Department of Physical Chemistry and Industrial NMR Centre,Stockholm, Sweden

What is this story about?• Phenomenon of internal magnetic fields (IMF) in

“porous solid / liquid” system studied by NMR: Subject, common problem and particular benefits

• Approach to study morphology of porous solid: Numerical calculations and NMR-experiment of Carr-Parcell-Meiboom-Gill (CPMG) echo trains for liquid diffusing in submicron pores

Phenomenon of IMF and NMR:Magnetic Susceptibility

magnetic susceptibility

• Magnetization of material put in external magnetic field:

0M H

• Local magnetic field and “Field Offset” inside material:

0 0* 1 ,B H and B B

• Local resonance frequency and frequency offset: 0* * ,f B and f B

• ...in term of chemical shift: 0f B

Phenomenon of IMF and NMR:Magnetic susceptibility

*Reproduced from: J.Schenck, Med. Phys. 23, 815 (1998)

Phenomenon of IMF and NMR:IMF in “porous solid / liquid” system

0B

0B

0 10 20 30 40 50 60 70 80 90

-1

0

1 CH 3 ( CH

2 )

3 Cl

in bulk in sand,

averaged particle size 50 m

I ( t

) / I

( 0 )

Time, ms8 6 4 2 0 -2 -4 -6 -8

Frequency Offset, kHz

CH 3 ( CH

2 )

3 Cl

in bulk in sand,

averaged particle size 50 m

• Distribution of local magnetic fields B* and resonance frequencies f* inside a porous solid:

10 2* , * *

liquid solid

B B f B

• ...on one hand, it leads to dramatic decrease of effective transverse relaxation time:

12* *T f

• …that, in turn, applies additional limitation on the use of many NMR techniques and high-resolution NMR…

Phenomenon of IMF and NMR:IMF in “porous solid / liquid” system

• On the other hand, distribution of local magnetic fields B* depends on pore morphology

The functional of IMF distribution [N. Fatkullin, Sov. Phys. JETF 74, 833 (1992)]:

3

23

2

1

1* 0 exp2

** ,

, ,

.,

sampleV

kk

k

a a is shortest length sc

B r W d r

B rB r

r

with IMF correlation

a

l

le of the pr

ength

and oblem

• IMF effects were exploited to characterize micron porous solids [see, for instance, Y.-Q.Song, Concepts Magn. Reson. A. 18, 97 (2003), R.V. Archipov et al., Appl. Magn. Reson. 29, 481 (2005)]

Phenomenon of IMF and NMR:IMF in system “porous solid / liquid”

1. Since pure analytical approaches for solving the problem of IMF effects are not applicable for arbitrary “porous media / liquid” system [see, for instance, Y.-Q. Song, Concepts Magn. Reson. A 18A, 97 (2003), P.N. Sen et al. J. Appl. Phys. 86, 4548 (1999), N.Fatkullin, Sov. Phys. JETP 74, 833 (1992)], the numerical strategy should be developed to calculate IMF effects of NMR observables

2. Which NMR technique can be used for practical observation of IMF effects* predicted by numerical procedure mentioned above?

Aims of this study:

*…the detection of IMF effects in submicron (non-transparent) porous solids is extremely interesting problem because (1) it has not been actively studied yet and (2) it might extend frames of NMR methods to small pore sizes and implements well-known experimental approaches [J.Kärger and D.M.Ruthven, Diffusion in Zeolites (John Willey & Songs Inc., 1992)]

Simulations:Porous solid ”skeleton”

• Material: Controlled-pore glasses (CPG, Millipore Corp., Billerica, MA, USA)

• Initial SEM image of CPG 1878 with mean pore size d = 1878 Å

• ...background formed by “ball” model• …the image of surface slice • …image of surface slice after binarization • ...calculated magnetic susceptibility distribution in 2D CPG model

MATLAB Imag

e

Process

ing Toolbox

Used

Simulations: Model

ijr

' 'i jr

,ij ijv

' ' ' ',i j i jv 0B

• consider voxel vij with local susceptibility χij (χij = const) in external field B0;

• the local magnetic field B*i’j’:

2' ' 0

0' ' 3

, ' ''

` '

3cos 1*

4

ij i ij j ij

i ji j ii jj

i ij j

r B vBB

r

▼ for protons in external field B0=11.7 T

Simulations: Algorithm

vi’j’,

ijr

' 'i jr

ij

' 'i j

vij, +0B

Мi’j’

B*i’j’

Mij

ij i jk ' '

B*ij

k k

kk

k k

solid

• diffusion was modeled by exchange of magnetization between voxels ij and i’j’ with probability kij↔i’j’ depending on the liquid self-diffusivity D0 [P.S. Belton et al., Mol. Phys. 61, 999 (1987)] and the geometry of porous media;

• algorithm is based on a stepwise calculation of the time dependence of the NMR signal [C. Mayer, Prog. Nucl. Magn. Reson. Spectrosc. 40, 307 (2002); C. Mayer, J. Chem. Phys. 118, 2775 (2003)]:

' '

*' '

,in

' '

*' '' '

:

e

e

ij

i j

ij iji B t

ijij i jfor all i jliquid phasei i j j

i B ti ji j ij

Local magnetization M t t M t

k tM t

k tM t

,

: iji j

Total magnetization M t M t

Simulations: Typical results for one 90° rf-pulse sequence

-1 0 1 2 3 4 5 6 7 8 9 10 11 12-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

NM

R s

igna

l

time, ms

Re (F.I.D.) Im (F.I.D.) | F.I.D. |

/ 2

Simulations: one 90° rf-pulse sequence

Comparison of the simulation results with data published early showed reasonable agreements:1. For motional averaging regime, the simulations give Δf1/2 ~ B0

2 [P.P.Mitra et al., Phys. Rev. B 44, 12035 (1991), N.F.Fatkullin, Sov. Phys. JETP 74, 833 (1992), P.N.Sen et al., J. Chem. Phys. 111, 6548 (1999), L.J.Zielinski et al. J. Magn. Reson. 147, 95 (2000), S.Axelrod et al. J. Chem. Phys. 114, 6878 (2001)]

2. The distribution of magnetization M(x,y) calculated for short time (not averaged) regime corresponds to pore morphology [Y.-Q.Song, Phys. Rev. Lett. 85, 3878 (2000), N.V.Lisitza et al., J. Chem. Phys. 114, 9120 (2001), Y.-Q.Song, Concepts Magn. Reson. A 18A, 97 (2003)]

100 101

100

101

102

FWH

M,

f 1/2 ,

Hz

B 0 , T

CPG with mean pore diameter d = 127.3 nm d = 187.8 nm d = 81.6 nm

f 1/2

~ B 20

…to Carr-Parcell Meiboom-Gill sequence• FID depends on many effects of spin dynamics (beside Zeeman

interactions):ˆ ˆ ˆ ˆ ˆ ˆ ˆ

Zeeman Quadrupolar Dipol Dipol rf J couplingH H H H H H H

• there are only few ways for studying diffusion exchange in system when parameters (first of all the magnetic susceptibility of components) – excluding diffusion – are held to be constants.

• Carr-Parcell Meiboom-Gill (CPMG) pulse sequence is well-known method to investigate exchange processes. For one small spin ensemble, IMF effects and their influence on CPMG decay shape may be expressed by:

2

.

1 1212

.

exp

exp 2 * *E E

E E

EE

Term I Effective transverse relaxivityof liquid inside the porous solid

n t n t

nt n tn

Term II Contribution to total relaxation process due todif

ntm tT

i dt f s t dt f s t

fusion of spins in IMF

CPMG echo-train: Experiment and simulation

Experiment: Materials• Porous solid: CPG1878, mean pore

diameter d=1878Å, magnetic susceptibility (at T=298K) χCPG= -(17.3±0.4)·10-6 [SI] ;

• Liquid at T=298K: buffer with pH=-13 χl= -(8.3±0.1)·10-6 [SI]; D0=(2.0±0.1) ·10-9 m2 s-1; T*2=(13.1±0.3)ms; T1=(2.83±0.01)s

• Sample: 98% pore space was filled by liquid, |Δχ|=(9.0±0.4)·10-6 [SI]Experiment: CPMG

• Spectrometer: AVANCE 500 B0=11.74 T, π-pulse was 7.62 μs and γB1/2π = 10.31kHz, hard pulse condition, |Δχ|B0<B1, was held.

• Echo-time delay, tE, was varied from 20 to 1320 μs

Simulations• Porous matrix: CPG1878, magnetic

susceptibility χCPG= -(17.3±0.4)·10-6 [SI]; binary 2D model of CPG was cut into voxels with side size l=10.236 nm (~18 cells per pore)

• Diffusant: ‘buffer-13’ χl= -(8.3±0.1)·10-6 [SI]; D0=(2.0±0.1) ·10-9 m2 s-1;

• Exchange constant k: k=D0l-2Fij↔i’j’ , where Fij↔i’j’ is 1 if cell (i’j’) is filled by liquid otherwise Fij↔i’j’ equals 0.

• Time step Δt = 3 ns.• π-pulse effect was simulated by complex

conjugation of transverse magnetization at the time of the pulse.

• Programs were written in MATLAB 7.0 and Fortran 95.

• We used typical desktop (Pentium D 2.7GHz, RAM 2Gb) and laptop (Centrino 2GHz, RAM 1.5Gb)

CPMG echo-train:Experiment and simulation

0 250 500 750

10-3

10-2

10-1

100

e-3

e-2

Nor

mal

ized

Spi

n-E

cho

Am

plitu

de,

M (

t ) /

M (

0 )

t, ms

e-1

Experimental CPMG Simulated CPMG

tE = 20 μs

tE = 132 μs

tE = 700 μs

tE = 1 ms

tE = 1.32 ms

0 100 200 300 400 500

10-2

10-1

100

e-3

e-2

Nor

mal

ized

Spi

n-E

cho

Am

plitu

de,

M (

t ) /

M (

0 )

t, ms

1234

e-1

tE = 20 μstE = 27.2 μs

tE = 81.4 μs

tE = 438 μs

The quantity evaluating the exponentiality of CPMG decays is function Ε(tE)

21 22 , ,2 0 0

e eeE

e

M T M TTt with e eT M M

Ε(tE)=1 for exponential decay and Ε(tE)≠1 for non-exponential decay

CPMG echo-train:Experiment and simulation

Experimental E-functions Simulated E-functions

21 22 , ,2 0 0

e eeE

e

M T M TTt with e eT M M

0

100

200

T eN

( t E

) , m

s ( a )

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.00

1.05

1.10 ( b )

N (

t E )

t*E

t E, ms

0

100

200( a )

T eN (

t E )

, ms

0.0 0.1 0.2 0.3 0.4 0.5

1.00

1.05

1.10

t Е , ms

( b )

N (

t E )t*E

Te Te2 Te Te2

CPMG decay shape

2

.

1 1212

.

exp

exp 2 * *E E

E E

EE

Term I Effective transverse relaxivityof liquid inside the porous solid

n t n t

nt n tn

Term II Contribution to total relaxation process due todif

ntm tT

i dt f s t dt f s t

fusion of spins in IMF

Regime Features of spin mobility for time-delay tE

Shape of CPMG-decay

Short tE

Spins diffuse inside small volume(s) where local MF is close to be constant

Decay’s shape is close to exponential one; contribution of

Term (II) is small.

Inter-mediate

tE

Spins cross few volumes differing in local MF; at the logging of NMR signal, each small spin ensemble is characterized by individual IMF effect; there is set of values

CPMG-decay is not exponential; contribution of

Term (II) is growing

Long tE

Spins cross many volumes differing in local MF; at the logging moment, all small spin ensembles are characterized by an averaged value

CPMG-decay is close to exponential; contribution of

Term (II) increases

1 12

12

' * ' ' * 'E E

E E

n t n t

nt n tdt f s t dt f s t

12 ' * 'E

E

n t

ntdt f s t

12 ' * 'E

E

n t

ntdt f s t const

Ε(tE) functions:Motional averaging scale, relation to pore

morphology

1 2 10

0.95

1.00

1.05

1.10

N (

t E )

( D 0 t E ) 1 / 2 / d

0

Experiment

Simulation

1. The simulation results generally reproduced experimental data

2. -function maxima position corresponds to two CPG1878 averaged pore diameter; it can be explained by elongated pore morphology of the CPG media [O.Petrov et al., Phys. Rev. E 73, 011608 (2006)] and presence of cavities at the channel cross points where effective pore size is obviously larger then channel diameter d0.

Summary1. The CPMG experiments performed for the controlled-pore glasses with

1878Å averaged pore diameter filled by aqueous 13 pH buffer showed resuts similar to to data of [L.J.Zielinski et al., J. Magn. Reson. 147, 95 (2000); S.Axelrod et al. J. Chem. Phys. 114, 6878 (2001); G.Q.Zhang et al. J. Magn. Reson. 163, 81 (2003); T.M. de Swiet et al. J. Chem. Phys. 100, 5597 (1994)], three relaxation regimes can be conventionally recognized with respect to the echo-time duration: short echo-time (similar to free-diffusion limit), intermediate (similar to localization) and long echo-time (motional averaged) regimes.

2. It was shown that shape of CPMG spin-echo attenuation evolves from exponential at short echo-times to non-exponential one at intermediate tE, and it becomes again roughly exponential at long echo-times. The CPMG-decays have the most pronounced non-exponential shape when spins diffuse over distance of two CPG pore diameters. The last result shows that study of the echo-time dependence of CPMG decay shape accompanied with appropriate computer calculations can provide information about morphology of some submicron porous solids that is not available for typical pulsed field NMR and MRI techniques, even at high external magnetic field.

Acknowledgements • The author thanks all of you for attention! • Knut and Alice Wallenberg Foundation and the

Swedish Research Council VR are acknowledged