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KJM-MENA4010 - Eksperimentelle metoder
Module: Diffusjon av væsker innesluttet i porøse materialer ved bruk av NMR.
Kort om emnet
Emnet gir en praktisk (og teoretisk) innføring i hvordan bestemme diffusjon av væsker
innesluttet i porøse materialer med NMR.
Forkunnskaper: Basiskunnskap i matematikk og kjemi
Tid: Vårsemesteret
Hva lærer du?
Studenten skal utføre diffusjon- og relaksasjonstidsmålinger på vann innesluttet i et
porøst materiale med et lav-felt NMR instrument. Med bakgrunn i enkel teori skal
overflatearealet i det porøse materialet bestemmes og graden av vekselvirkning mellom
vannet og den porøse overflaten diskuteres.
KJM-MENA4010 - Eksperimentelle metoder
Module: Diffusion of Fluids within Porous Materials Probed by NMR.
(Water confined between glass beads)
Eddy W. Hansen
UiO, February 2013
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Introduction
Nuclear Magnetic Resonance (NMR) represents a notable tool in characterizing
molecular dynamics within both solids and liquids. In general, NMR is capable of
pinning down the motional characteristics (correlation time) on a molecular level
of any fluid confined in a porous matrix, like catalytic materials (zeolites and
mesoporous materials), sandstone, clay, wood, plants and biological materials,
i.e., the human body. The transport properties of a pore confined fluid are
generally controlled by:




The temperature and pressure
The properties of the confined fluid (molecular dimension and type of
fluid-fluid interaction)
The properties of the porous matrix (surface-to-volume-ratio or pore
dimension and pore connectivity)
The interaction strength between the fluid and the surface of the porous
matrix (surface relaxivity).
Hence, NMR should be of interest to anyone interested in material science
(physics, chemistry, biology, catalysis and medicine).
In this project we will measure the spin-lattice relaxation time (T1) and the
diffusivity (D) of pore confined water with the objective to estimate a) the surfaceto-volume (S/V) ratio of the porous matrix and b) the surface relaxivity (of
water.
Although a large fraction of the nuclei in the periodic table are NMR active, only
the proton nucleus will be the probe molecule in the present work. The outlined
theory will, however, be of general validity.
Activities
The module starts with a lecture (approximately 2 - 4 hours), laboratory work
(under supervision) and data-analysis. The student submits a written report at the
end of the course.
Teaching material
Distributed by e-mail or at the first lecture
Time schedule
2 weeks. We start with an introductory meeting. More information will be given
during the meeting.
Objects
 To understand the concept of relaxation and diffusion from a
phenomenological point of view (Bloch equations)
 To be able to perform relaxation and diffusion measurements
 To derive information on diffusivity, surface-to-volume-ratio and surfacefluid interaction by model fitting.
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

To understand the additional complexity involved in analyzing NMR data
from pore-confined fluids, as compared to bulk fluids.
To recognize the applicability of NMR in various aspects of material science.
Theory1
As pointed out in the introductory section, most of the nuclei in the periodic table
possess a magnetic moment (). For a proton nucleus, the magnetic moment
along the external field B0 (defining the z-axis) will be either positive (+z) or
negative (z). At thermal equilibrium, a surplus of the proton nuclei within a
sample will have their magnetic moments oriented along the magnetic field
direction B0 and hence generate a macroscopic magnetic moment M0 aligned
along this direction, as illustrated in Fig.1A.
B0
z
A)
z
B)

coil
B1
y
x
y
x
Figure 1 A) At equilibrium the macroscopic magnetization M0 is aligned parallel to the static
magnetic field B0. B) If applying a radio frequency (rf)-pulse B1 of duration t along the x-axis, the
macroscopic magnetization will rotate an angle B1t around this axis.
Since the detector coil (y-axis) is aligned normal to the external magnetic field B0,
no signal will be detected (Figure 1A) since the macroscopic magnetization has
no component along this axis. In order to detect a signal the magnetization must
be tilted away from the z-axis, as illustrated in Figure 1B. Such a reorientation or
rotation of the magnetization is performed by irradiating the sample with a radio
frequency (rf) pulse of duration tp having a magbetic component of magnitude B1.
The resulting signal observed after applying such a rf-pulse is illustrated on
Figure 2.
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Figure 2. NMR signal intensity (detected in the receiver coil, see Figure 1B) against time as
observed after applying an rf-pulse of duration 2.15 s, corresponding to a tip-angle of 900. In
short, the macroscopic magnetization is rotated 900 around the x-axis and onto the y-axis).
The Bloch equation
(Phenomenological description of the time dependent magnetization)
How the macroscopic magnetization ( M  M x i  M y j  M z k ) develops in time can
be described phenomenologically by solving the Bloch equation;
M
M
M  M0
M
  M xB  x i  y j  z
k  D 2 M
t
T2
T2
T1
(1)
Where  2   2 / x   2 / y   2 / z , B represents the magnetic field, D is the
self-diffusion coefficient, T1 is the spin-lattice relaxation time and T2 defines the
spin-spin relaxation time of the fluid, respectively. This vector equation reveals
two independent signals to appear in the NMR experiment; a transversal
magnetization component Mx(y) decaying to zero with a rate constant 1/T2 and a
longitudinal magnetization component Mz(t) which increases asymptotically
towards the equilibrium value M0 with a rate constant 1/T1. For a pore confined
fluid, the spin-spin relaxation time T2 shows up to be a somewhat difficult
parameter to analyze (although possible). Hence, in this project, only T1
measurements will be implemented and discussed.
Spin-lattice relaxation time (T1)
Measurement
If applying an rf-pulse which inverts (-pulse) the longitudinal magnetization, i.e.,
Mz = -M0, the time behavior of this longitudinal magnetization Mz(t) can easily be
determined by simple integration of Eq 1 with respect to time (excluding the
diffusion term in Eq 1);
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Mz

M 0
dM ' z

M 0  M 'z
t

0
dt
T1


M z  M 0 1  2e  t / T1

(2)
The experimental approach to determine T 1 is thus to apply the following pulse
sequence;
Figure 3. Schematic view of the inversion recovery pulse sequence used to measure T 1.
in which the longitudinal magnetization is measured at various times t = ti. For a
given ti, the pulse sequence is repeated a finite number of times (number of
scans) in order to improve the S/N-noise ratio (Figure 5). The time delay tR
between each inversion sequence should be chosen so that t R > 5*T1.
Figure 4. A series of inversion recovery pulse sequences, with a time delay t R
between the read pulse (/2-pulse) in one sequence and the inversion pulse in
the next sequence.
Pore Model
We will assume the fluid molecules to be confined in a pore of volume V (= Vs +
Vb) and surface area S (Figure 5) where Vs and Vb are the pore surface- and
pore volume, respectively. The fluid molecules on the pore surface are
characterized by a surface spin-lattice relaxation rate 1/T1s, and the bulk fluid
molecules are characterized by a corresponding bulk spin-lattice relaxation rate
1/T1b.
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Vs, d, 1/T1s
Vb, 1/T1b
Figure 5. Illustration of a random pore geometry of volume V(=Vs + Vb) and
surface area S. The thickness of the surface fluid layer is d.
If assuming that the fluid molecules on the pore surface (volume VS; blue area,
Figure 5) and within the pore (volume Vb; white area, Figure 5) are exchanging
fast on the NMR time scale, we find that the spin-lattice relaxation rate 1/T1 of the
fluid molecules can be represented by:
V 1
1 Vb 1

 s
T1 V T1b V T1s
(3a)
where the density of the water molecules (= the number of molecules per unit
volume) is assumed to be the same for both the surface and the bulk regions,
respectively. Moreover, since VS ≈ d.S and V = Vb + Vs we can rewrite Eq 3a to
read;
1
1
d S 1
1


(

)
T1 T1b
V T1s T1b
(3b)
If T1s << T1b (which is often the case when the interaction between the matrix and
the surface molecules is significant), Eq 3b simplifies to;
1
1
d S 1
1
S




T1 T1b
V T1s T1b
V
(3c)
where the parameter  (= d/T1s) defines the surface relaxivity. The surface
relaxivity for some fluid-matrix systems are shown in Table 1.
Table 1. Catalog of surface relaxivity r for water saturated unconsolidated materials with welldefined surface area from BET or grain size2
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Self-diffusion
The simplest pulse sequence to consider when aiming at measuring the self
diffusion of a fluid is illustrated in Figure 6.
Figure 6. Illustration of the simplest pulsed field gradient scheme used to measure diffusivity. g
represents the strength of the gradient field of duration . The timing of both rf-pulses and
gradient pulses are illustrated along the time axis.
From basic NMR theory the following equation can be derived (the interested
reader is referred to the Appendix A);

M
 K  exp   2 2g 2 (   / 3)D
M0

(4)
whereis a constant denoted the gyromagnetic ratio (= 2.675.108 rads-1), M is
the macroscopic magnetization, g is the gradient field strength and D is the
diffusivity. The parameters  and  define the time duration of the gradient pulse
and the time distance between the /2- and the - rf-pulse (see Figure 6). For a
specific system and a certain , the factor K in Eq 4 can be considered a constant
When measuring the diffusivity of fluid molecules confined in porous materials
the effect of internal gradients, eddy currents (caused by strong gradient pulses),
unwanted echoes (when applying more than a single rf-pulse) and the
observation that T2 is frequently much shorter than T1 must be considered. This
calls for designing a pulse sequence of a significant more complexity, denoted
the “13-interval pulse sequence”, and is illustrated in Figure 7. It is beyond the
scope of this project to detail the derivation of this formula, however, the
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approach is the same as for deriving Eq A4 (Appendix). We simply present the
result (Eq 5) without further discussion;
Figure 7. Illustration of the “13 interval pulse gradient stimulated echo sequence” using bipolar
gradients. g represents the strength of the gradient pulse of duration . The timing of both rfpulses and gradient pulses are illustrated along the time axis.

M
 K '  exp  4 2 2 g 2 D(t D )t D
M0

(5)
t D    3 / 2   / 6
Eqs 4 and 5 show that D of a confined fluid depends on the diffusion time t D. In
contrast, for bulk solutions D is independent of tD. This can be rationalized by the
spatially restricted diffusion of the pore confined fluid, since an increasing fraction
of the molecules will sense the restriction as the diffusion progresses. Hence, we
expect the effective diffusivity D to decrease with increasing diffusion time, as
reflected by Eq 5.
For long diffusion times, the diffusivity will approach a limiting value,
corresponding to the long-range diffusivity limit.
Using general physical arguments it can be further shown that for short diffusion
times the following Equation is valid3.
D(t D )
4 S
 1
D(0)
9  V
(6)
D(0)t D
Which can be rearranged to read:
Y  aX  b with X  t D , Y  D(t D ), b  D(0), and a  
4
9 
b3 / 2
S
V
(7)
In order to derive a and b, we simply plot Y against X and apply a linear fit.
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PROJECT
a) Measure T1 of a) bulk water and b) water confined between glass beads
(porous model system). Discuss briefly why these relaxation times are different.
b) Measure the diffusivity D(tD) of pore confined water at different diffusion times tD
and plot D(t D ) against
t D (in which tD has dimension s and D has dimension
cm2s-1). Fit a straight line (Eq P1) to the experimental data and determine the
bulk diffusivity D(0) and the surface-to-volume ratio S/V.
D( t D )   
4
9 
D(0)3 / 2
S
t D  D(0)
V
(P1)
c) Show that the S/V-ratio for a spherical pore with radius R can be written:
S/V = 3/R
(P2)
and estimate R.
d) You have measured the spin-lattice relaxation time T1b of bulk water and the
corresponding T1 of water confined between glass beads, T1, water within glassbeads.
Estimate the surface relaxivityfrom Eq P3:
1
T1 ,water
in glassbeads)

1 d S 1
1
S



T1b
V T1s T1b
V
(P3)
Compare your result with the data presented in Table 1.
f) Write a short report on your work. The report should be sectioned according to:
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




Introduction (what you want to do and why)
Theory (short description of the theory, including the basic equations you
apply).
Experimental
Results (Figures showing experimental raw data)
Discussion (answer the questions a – d). Try to estimate uncertainties in
derived parameters (use Origin!).
APPENDIX A. For those students who want a more profound and deeper
theoretical insight into the Pulse-field gradient (PFG) NMR technique.
It can be shown that for fluids undergoing diffusion in a gradient field (g), the
Bloch equation takes the form;
M 
M
 izgM  
 2M 
t
T2
with;
M   M x  iM y
(A1)
After some complex and elaborate calculations4 the solution to Eq A1 can be
written;
t t'


M   M  exp  t1 / T2   2 D  (  g (t ' ' )dt ' ' ) 2 dt '
0


0 0


(A2)
where D is the diffusion coefficient, g(t΄΄) is the time dependent magnetic field
(gradient pulses), T2 is the spin-spin relaxation time and t1 represents the
duration when the NMR signal is influenced by transverse relaxation processes.
We will apply Eq A2 to the following situation;
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Figure A1. Illustration of the simplest pulsed field gradient pulse sequence used
to measure diffusivity. g represents the strength of the gradient field of duration .
The timing of both rf-pulses and gradient pulses is shown along the time axis.
t'
We can illustrate the time behaviour of the phase integral  g (t ' ' )dt ' ' within the
0
respective time intervals:
g(t΄)
g
t΄


2
-g
t'
Figure A2. Illustration of the time dependence of the phase angle:
 g (t ' ' )dt ' ' .
0
We can then easily calculate the integral representing the phase change (see Eq
A2) to obtain:
Table. Effective gradients and integrals1)
Time interval i
1
2
Time interval
t1=0 t2=
t2= t3=
Effective
Gradient
g(t’’)
  g i (t ' )   g (t ' ' )dt ' ' (  g (t ' ' )dt ' ' ) 2
0
g
0
0
gt   g (   ) / 2
g 2 (t '2  (   )t '
t'
t'
0
0
1 / 4  (   )2 )
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3
t3 t4=
0
g
4
t4 t5=3/
0
- g
g 2 2
5
t5 t6=3/
g
 g  g (t   (3   ) / 2)
=gt΄-g(3
g 2 (t '2  (3   )t '
g 2 2
1 / 4  (3   ) 2 )
6
0
0
t6 t7=2
0
1) Since we are only considering linear gradients (no pulse-shaping) the phase angle (column 4) can
always be written   t    , where  = 0 (no gradient pulse applied) and  = g or  =g. The
parameteris determined by setting  at the end of one time interval equal to  at the start of the
subsequent time interval. One exception is the time t4 at which the phase is inverted due to the rf-pulse.
2
t'
 ( g (t ' ' )dt ' ' ) dt ' 
2
0



t'
(  g (t ' ' )dt ' ' ) dt '
2
t'
 / 2  / 2 0
3 / 2  / 2
0
3 / 2  / 2
0
3 / 2  / 2
 / 2  / 2
(  g (t ' ' )dt ' ' ) 2 dt '



 / 2  / 2
t'
(  g (t ' ' )dt ' ' ) 2 dt '
0
t'
(  g (t ' ' )dt ' ' ) 2 dt '
0
(t '3 / 3  (   )t '2 / 2  (   ) 2 t ' / 4) (  ) / 2  ( 2t ' )
(   ) / 2
 g 
 (t '3 / 3  (3   )  t '2 / 2  (3   ) 2 t , / 4) ( 3  ) / 2
( 3  ) 2

2 2
  g (   / 3) D
2
(A3)
( 3  ) / 2
(   ) / 2




Hence:

I
 exp   2 g 2 (   / 3) D
I0

(A4)
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Reference
1. Jiansheng Chen, NMR Surface Relaxation, Wettability and OBM Drilling Fluids, PhDthesis work, Rice University, Houston Texas, 2005, 187 pages.
2. Hornak, Joseph P.: The Basics of NMR. http://www.cis.rit.edu/htbooks/nmr/nmrmain.htm A book with animations with a lot of literature references.
3. P.P.Mitra, P.N.Sen, L.M.Schwartz, Phys. Rev. B, 1993, 47, 8565.
4. R.F. Karlicek and I.J. Lowe, J. Magn. Res., 1980, 37, 75 and R.M. Cotts, M.J.R. Hoch,
T. Sun and J.T. Markert, J. Magn. Res., 1989, 83, 252.
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Practical NMR
A Simplified User Manual for the Maran Ultra NMR Instrument (Non-expert User)
(Teacher checks that no data are stored in the files GR1 – GR5 before start)
Initiate experiment
1. Insert a glass-beads-water sample in correct position
2. Acquisition mode: sequence → load → FID.EXE
3. Set RD to ≈ 5.T1, i.e., RD = 7.0s, P90 = 2.15 (s), VT = 35 (0C) and RG 2
Optimizing relevant parameters
4. Command → Auto O1 (wait)
5. Command → Auto RG (wait). Use this value of RG throughout if not
otherwise stated in the text.
6. Command → Auto P90 (wait)
T1 measurement on bulk water
7. Insert a bulk water sample in correct position
8. Acquisition mode → Sequence → Load → INVREC
9. Set RD 17.5s, RG 2, VT 35 and NS 4
10. Write: .T1 → T1-bulk-water → open → OK→ write a file name, for instance;
KJM-MENA4010/GR1/T1-B1). If you belong to a group i (≠ 1), then replace
GR1 by GRi.
11. A T1-plot will appear on the screen when the experiment is finished. Make a
“screen dump” (Press “Print Scrn” and paste the image into “Paint”). You will use
the value of T1 in the report.
T1 measurement of water confined between glass beads
12. Insert the glass-beads-water sample in correct position
13. Acquisition mode → Sequence → Load → INVREC
14. Set RD 7.0s, RG 2, VT 35 and NS 4
15. Write: .T1 → T1-glassbeads → open → OK→ write a file name, example; KJMMENA4010/GR1/T1-1. If you belong to group i (≠ 1), replace GR1 by GRi.
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16. A T1-plot will appear on the screen when the experiment is finished. Make a
“screen dump” (Press “Print Scrn” and paste the image into Paint). You will use
the value of T1 in the report.
(Misellaneous)
Calculations using the NMR program WinFit
17. Open WinFit: C:\ProgramFiles\Resonance\WinFit\nmrfit (double click)
18. Retrieve a file: File → Open → Select file → data → (Select Fit Options)
→number of exponents/DC-offset (if possible) →Auto Initialize → Fit → Data
(Read out values) or print, i.e.; File → Print → ….
19. You may also read the T1 data into an excel spread sheet;
Enter excel → File → Open → File name (*.INT)
Diffusion measurement
Initiating measurements
1. “Acquisition mode” → Sequence → Load → CPMGB13.EXE → Open
2. Set: SI 512, DW 0.1, NS 8, RD 7.0s, DEAD1 5, DEAD2 5, D1 250, D2 600,
D3 1000, D7 100, G1 100, G2 -100, G3 100, RG 6, C1 4, C2 2, FW 1E6. P90 2.1
3. Command → Auto O1
4. Command → Auto RG
5. Command → Auto P90
Start measurement
Retrieve the batch file .KJM-MENA4010-C20-1
If you belong to a group i ≠ 1 you simply replace all GR1 by GRi in the batch file
and store the new batch file as: KJM-MENA4010-C20-i
Start the experiment: .KJM-MENA4010-C20-i
The whole experiment lasts for approximately 3 hours.
Note:
6. KJM-MENA4010-C20-1 is a “batch-file” and enables all 8 diffusion
measurements (D7: = 500, 10000, 20000, 30000, 40000, 50000, 60000 and 70000
s) to be performed in one “run”.
7. Remember to use your correct group number GRi (i =1, 2, 3, 4 or 5) in the batchfile, else you may destroy data for other group members. Your data will be stored
in excel spread sheets (C:\KJM-MENA4010\GRi\D-xxxx; where xxxxx
represents the apparent diffusion time D7 in microseconds).
8. Transfer or copy your data for subsequent post-processing.
9. The actual data analysis can be performed using Excel (the supervisor will
prepare an Excel spread-sheet for data processing and data analysis).
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Appendix. “Batch-file” and gradient lists (f7_list and g7_list) used for
acquisition of diffusion data
Batchfile: FYS-KJM4010-C20-1
~AMODE
vt 35
load cpmgb13
C1 4
C2 2
d1 250
d2 600
rg 6
ns 8
d3 1000
dw 0.1
rd 7.0s
SI 512
G3 100
~AMODE
D7 500
.PFG_DIFF_AT_EWH f7_list g7_list
wr c:\KJM-MENA4010\GR1\D-500
EX c:\KJM-MENA4010\GR1\D-500
~AMODE
D7 10000
.PFG_DIFF_AT_EWH f7_list g7_list
wr c:\KJM-MENA4010\GR1\D-10000
EX c:\KJM-MENA4010\GR1\D-10000
~AMODE
D7 20000
.PFG_DIFF_AT_EWH f7_list g7_list
wr c:\KJM-MENA4010\GR1\D-20000
EX c:\KJM-MENA4010\GR1\D-20000
~AMODE
D7 30000
.PFG_DIFF_AT_EWH f7_list g7_list
wr c:\KJM-MENA4010\GR1\D-30000
EX c:\KJM-MENA4010\GR1\D-30000
~AMODE
D7 40000
.PFG_DIFF_AT_EWH f7_list g7_list
wr c:\KJM-MENA4010\GR1\D-40000
EX c:\KJM-MENA4010\GR1\D-40000
~AMODE
D7 50000
.PFG_DIFF_AT_EWH f7_list g7_list
wr c:\KJM-MENA4010\GR1\D-50000
EX c:\KJM-MENA4010\GR1\D-50000
~AMODE
D7 60000
.PFG_DIFF_AT_EWH f7_list g7_list
wr c:\KJM-MENA4010\GR1\D-60000
EX c:\KJM-MENA4010\GR1\D-60000
~AMODE
D7 70000
.PFG_DIFF_AT_EWH f7_list g7_list
wr c:\KJM-MENA4010\GR1\D-70000
EX c:\KJM-MENA4010\GR1\D-70000
Gradient lists
16
D:\533564841.doc
f7_list
600
1200
1800
2400
3000
3600
4200
4800
5400
6000
6600
7200
7800
8400
9000
9600
10200
10800
11400
12000
g7_list
-600
-1200
-1800
-2400
-3000
-3600
-4200
-4800
-5400
-6000
-6600
-7200
-7800
-8400
-9000
-9600
-10200
-10800
-11400
-12000
17