Nuclear magnetic resonance studies of structure
and dynamics in heterogeneous samples
by
Gabriela Leu
B.S. Physics, Al.I.Cuza University (1993)
Submitted to the Department of Nuclear Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2003
c
°Massachusetts
Institute of Technology 2003. All Rights Reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Nuclear Engineering
September, 2003
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David G. Cory
Professor, Department of Nuclear Engineering at MIT
Thesis Supervisor
Read by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sow-Hsin Chen
Professor, Department of Nuclear Engineering at MIT
Thesis Reader
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jeffrey A. Coderre
Associate Professor and Chairperson, Nuclear Engineering
Department Committee on Graduate Students
2
Nuclear magnetic resonance studies of structure and
dynamics in heterogeneous samples
by
Gabriela Leu
Submitted to the Department of Nuclear Engineering
on September, 2003, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
The aim of this work is to develop and implement methods for determining the local
structure and dynamics of heterogeneous samples (e.g. rocks, rubber, porous materials, etc.). From the physical point of view, the heterogeneities are best described
in terms of variations in the local susceptibility. The proposed methods are based
on analyzing the modulation of spin dynamics by the Magic Angle Sample Spinning
(MASS) method in the presence of both local variations in the bulk magnetic susceptibility and molecular diffusion. The correlations between the dipolar and susceptibility
fields are used for extracting information on the detailed structure and composition of
heterogeneous samples. In the first part of this dissertation, a new method for characterizing porous media, diffusive MASS, is presented. This method is combining
MASS and molecular diffusion, providing thus a unique way to simultaneously obtain high resolution spectra and information on the geometry and the internal fields
inherent to porous samples. The second part is concerned with obtaining detailed
information about the structure and dynamics in rubber samples with the aim of understanding the elastomer-carbon black interactions. The processes that occur at the
elastomer-carbon black interface are responsible for the special properties that make
rubber so useful and these processes are not yet fully understood. We use the dipolar and susceptibility interactions to characterize the elastomer spatial distribution
relative to the carbon black surface, the elastomer mobility and the local order. The
last section presents a MASS study of the relaxation and wettability of actual rock
samples. It is important to design experiments for characterizing wettability which
are less time consuming than the current core-flooding and imbibition experiments.
The combination of MASS and relaxation measurements permits the determination
of the chemical composition and wetting fluid in core samples. We apply this method
to the characterization of two preserved sandstone cores.
Thesis Supervisor: David G. Cory
Title: Professor, Department of Nuclear Engineering at MIT
Acknowledgments
I would first like to express my deep gratitude to my adviser, Prof. David Cory.
His sustained support, kindness and dedication as research adviser and enthusiastic
teacher, have guided and helped me through my Ph.D. studies.
My gratitude goes to Dr. Pabitra Sen, for being an invaluable, constant source of
help and inspiration for my research. His kind support and willingness to share his
scientific experience were invaluable for all my work.
I would like to thank the members of my committee Prof. Sow-Hsin Chen, Prof.
Sidney Yip and Prof. Jeffrey Coderre for their help and useful suggestions.
I am sincerely grateful to all the colleagues that I had over the years in the Cory
group for always willing to help and for making the lab a pleasant and inspiring work
place. In particular I would like to thank to Yun Liu with whom I collaborated closely.
I am also grateful to Dr. Werner Maas for introducing me to the practical aspects of
the NMR experiments.
I would also like to thank to my very best friend, Vio, for his support. He has
helped me through many difficult times, and his unwavering faith in me has always
been a steady source of confidence and motivation.
I will be forever grateful to my parents for encouraging me to follow this path
and for initiating and nurturing my interest in science. I can not thank them enough
for their support, understanding and patience during all these years. This thesis is
dedicated to my entire family.
6
Contents
1 Molecular diffusion effects on magic angle sample spinning
1.1
1.2
19
Amplitude modulation and relaxation due to diffusion in NMR experiments with a rotating sample . . . . . . . . . . . . . . . . . . . . . .
19
1.1.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.1.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.1.3
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.1.4
Experimental results and discussions . . . . . . . . . . . . . .
26
1.1.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Manipulation of phase and amplitude modulation of spin magnetization in magic angle spinning NMR in the presence of molecular diffusion 31
1.2.1
Partial suppression of sidebands with TOSS . . . . . . . . . .
1.2.2
Equation of motion of magnetization for a spinning sample in
32
the presence of diffusion . . . . . . . . . . . . . . . . . . . . .
34
1.2.3
MASS and TOSS without diffusion . . . . . . . . . . . . . . .
35
1.2.4
MASS and TOSS with diffusion . . . . . . . . . . . . . . . . .
37
1.2.5
Model phase factor . . . . . . . . . . . . . . . . . . . . . . . .
37
1.2.6
Diffusion in the presence of a single dipole . . . . . . . . . . .
38
1.2.7
Experimental results . . . . . . . . . . . . . . . . . . . . . . .
39
1.2.8
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2 Two-dimensional exchange diffusive magic angle sample spinning
49
2.1
Detection of motion though susceptibility fields in two-dimensional exchange diffusive-MASS experiments . . . . . . . . . . . . . . . . . . .
7
49
2.2
2.1.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.1.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.1.3
Two-dimensional exchange in diffusive MASS . . . . . . . . .
51
2.1.4
Experimental results and discussions . . . . . . . . . . . . . .
56
2.1.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Selectively observing the amplitude modulation under MASS . . . . .
61
2.2.1
Phase suppressed method . . . . . . . . . . . . . . . . . . . .
63
2.2.2
Two-dimensional exchange method . . . . . . . . . . . . . . .
66
2.2.3
Experimental results of the phase suppressed method . . . . .
67
2.2.4
Experimental results of the two-dimensional exchange method
70
2.2.5
Validity of the approximation for the phase suppressed method
73
2.2.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3 Two-dimensional NMR-DQF studies of heterogeneous samples: quantitative characterization of elastomer-carbon black interactions
79
3.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
3.2.1
Properties of rubber-like samples . . . . . . . . . . . . . . . .
82
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.3.1
Powder average . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.3.2
DQF dipolar/susceptibility spectroscopy . . . . . . . . . . . .
85
3.3.3
Components of two-dimensional DQF spectra . . . . . . . . .
87
3.3.4
Orientation of PAS between dipolar and susceptibility tensors
89
3.3.5
Simulations of the encapsulated and surface elastomers . . . .
89
3.3.6
Relative strength of the dipolar and susceptibility local fields .
90
3.3.7
Lengthscale estimation for the surface component . . . . . . .
91
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.4.1
Two-dimensional DQF measurements on rubber samples . . .
93
3.4.2
Relative strength of the dipolar and susceptibility local fields . 101
3.4.3
Relative amounts of elastomer components . . . . . . . . . . . 101
3.3
3.4
8
3.5
3.4.4
Estimation of CB particle sizes: SEM/TEM/AFM experiments 105
3.4.5
NMR susceptibility measurements of carbon black . . . . . . . 112
3.4.6
Characterization of the encapsulated elastomer . . . . . . . . . 112
3.4.7
Lengthscale estimation for the surface component . . . . . . . 116
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4 NMR identification of fluids and wettability in preserved cores in
situ
123
4.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3
X-ray data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.4
MASS data on rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.5
Inversion-recovery with MASS . . . . . . . . . . . . . . . . . . . . . . 129
4.6
CPMG with MASS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 General conclusions
139
9
10
List of Figures
1-1 Comparison of CPMG (static sample) and MASS (rotating sample)
attenuation data on a pack of beads as a function of
n
ωr3
(n is the echo
number, ωr = 2π/τr the rotor frequency. For purposes of comparison,
we express the static CPMG data in using 2τ = τr . Note that τr is the
time between two echoes for MASS and 2τ is the time between two
echoes for CPMG). The early times data show the “free diffusion” behavior. The solid lines correspond to ln(A/A0 )M ASS = − 14.9×10
ω3
r
ln(A/A0 )CP M G = −
7.5×1010
ωr3
n
10
n
and
. At later times the effects of restriction
and the breakdown of other approximations are apparent. . . . . . . .
27
1-2 The TOSS sequences that we used consists of four π pulses followed
by a waiting time, all chosen to satisfy TOSS conditions. . . . . . . .
36
1-3 Comparison of wax and water spectra, both embedded in a pack of
beads and subjected to the first TOSS sequence from Table 1.1. . . .
42
1-4 Comparison of the two different TOSS sequences for water in a pack
of glass beads (νr =1kHz). The thin line corresponds to the first TOSS
time sequence from Table 1.1 while the thick line corresponds to the
second sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2-1 Pulse sequence of 2D-exchange experiments. . . . . . . . . . . . . . .
51
2-2 MASS 2D exchange spectrum of wax embedded in a pack of beads
(tm =30ms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
2-3 MASS 2D exchange spectrum for water embedded in a pack of beads
(tm =5ms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
58
2-4 The 45o integration spectrum of 2D spectrum of wax. The insert is the
slice spectrum along the diagonal line of the 2D spectrum of wax. . .
58
2-5 The 45o integration spectrum of 2D spectrum of water in bead pack.
The insert is the slice spectrum along the diagonal line of the 2D spectrum of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
2-6 Results of phase suppressed experiments of water in packed 50µm glass
beads showing the dependence of the ensemble amplitude modulation
on t1 (Eq. 2.13) for three values of the mixing time (tm = 10ms, 100ms
and 1000ms), at a spinning frequency of 2kHz. The data for each
experiment was normalized with respect to the first data point. The
curves obtained by fitting the experimental data points with Eq. 2.19
are shown as solid lines. . . . . . . . . . . . . . . . . . . . . . . . . .
68
2-7 Increase in g1 with the diffusion distance in samples with different bead
sizes. All the samples are packed glass beads filled with 1% water and
99%D2 O. The sample corresponding to the middle curve is packed one
half with 50µm glass beads and one half with 100µm glass beads. The
spinning frequency in all the experiments is 5kHz. . . . . . . . . . . .
70
2-8 The Fourier Transform of the amplitude modulation for different mixing times obtained by two-dimensional exchange method in 50µm glass
beads sample filled with 1% water and 99% D2 O. The center peak of
each spectrum is normalized as one. The spinning frequency is 2kHz.
71
2-9 The normalized second moment RM2 (the ration of the second moment
of the amplitude modulation spectrum to the second moment of the
Fourier Transform of the FID) for different glass beads collapses when
plotted as a function of scaled diffusion length. . . . . . . . . . . . . .
72
2-10 The change of g1 with different spinning frequencies. . . . . . . . . . .
73
2-11 The change of g2 with different spinning frequencies. . . . . . . . . . .
74
3-1 Structural properties of carbon black particles [20]. . . . . . . . . . .
82
12
3-2 Polybutadiene (PBD) and the characteristic 1 H dipolar couplings (ωD =
µ0 γ 2
h̄ )
4π r 3
in both cis and trans configurations. . . . . . . . . . . . . . .
83
3-3 The NMR frequency depends on the relative molecular orientation with
respect to the external magnetic field (ω = ωD(χ) (1 − 3 cos2 θ)). By
integrating over all possible orientations, the dipolar and respective
susceptibility lineshapes are obtained. . . . . . . . . . . . . . . . . . .
85
3-4 Double quantum filtering pulse sequence. . . . . . . . . . . . . . . . .
86
3-5 Typical 2-D DQF spectral correlation map. The three distinct spectral
regions used in our analysis are emphasized (dotted lines). . . . . . .
87
3-6 Simulation results showing the two-dimensional spectrum obtained for
different orientations of the PAS between the dipolar and susceptibility
tensors. The simulation with 0o between the PAS of the dipolar and
susceptibility tensors is in a very good agreement with the experimental
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
3-7 Simulation results showing the 1st (encapsulated) and 2nd (surface)
elastomer components (see text for details). The maximum intensity
peaks that define the 1st (I1 ) and 2nd (I2 ) component are shown. . . .
90
3-8 Simulations results showing the variation of the angle θ between the
most intense peaks corresponding to the 2nd component for different
relative strength of the dipolar and susceptibility fields. . . . . . . . .
3-9 The variation of θ with for
ωD
.
ωχ
. . . . . . . . . . . . . . . . . . . . . .
91
92
3-10 Simulation results showing the field distribution in spherical shells of
different thickness around the carbon black particle. . . . . . . . . . .
94
3-11 The dependency of the 2nd moment of the field distribution as a function of the shell thickness. . . . . . . . . . . . . . . . . . . . . . . . .
95
3-12 Two-dimensional DQF spectrum obtained from the sample 5G. Only
the encapsulated (1st component) is present in this sample. . . . . . .
96
3-13 Two-dimensional DQF spectrum obtained from the sample SnBr. All
the three elastomer components are present. . . . . . . . . . . . . . .
13
97
3-14 Two-dimensional DQF spectrum from the sample 5G and the F1 and
F2 projections. The projections along both axes are the identical and
the spectral broadening is due to the very limited elastomer mobility
99
3-15 Two-dimensional DQF spectrum from the sample SnBr and the F1
and F2 projections. The broad dipolar component is due to the encapsulated and surface elastomer components discussed above and it
has similar lineshape with the F 1 projection from the 5G sample. The
narrow dipolar component is due to the 3rd elastomer component. Significant asymmetry observed in the F 2 projection.
. . . . . . . . . . 100
3-16 Two-dimensional DQF results for all the samples used in this study.
The angle θ between the most intense peaks corresponding to the 2nd
component is shown and it is about 1320 for all the samples. . . . . . 102
3-17 F1 projection of the SnBr sample. The 3rd component is fitted with a
Lorentzian lineshape and extracted. . . . . . . . . . . . . . . . . . . . 103
3-18 F2 projection of the SnBr sample. The 3rd component is fitted with an
axially asymmetric powder pattern. . . . . . . . . . . . . . . . . . . . 104
3-19 AFM image of the 5G sample. . . . . . . . . . . . . . . . . . . . . . . 105
3-20 AFM image of the SnBr sample. As expected, we observed that the
carbon black particles are less concentrated than in the 5G sample. . 106
3-21 SEM image of the carbon black powder. The mean diameter of the
carbon black particles is 35±4.5 nm. . . . . . . . . . . . . . . . . . . 107
3-22 SEM image of the carbon black powder at higher magnification. Artifacts due to the sample charging become visible. . . . . . . . . . . . . 108
3-23 SEM image of the SnBr sample. The mean diameter of the carbon
black particles is 110±17 nm. . . . . . . . . . . . . . . . . . . . . . . 108
3-24 SEM image of the 5G sample. The mean diameter of the carbon black
particles is 83±9.7 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3-25 TEM image of the carbon black powder. . . . . . . . . . . . . . . . . 110
3-26 Histograms showing the carbon black particle size distributions. The
raw data and the corresponding Gaussian distributions are shown. . . 111
14
3-27 Carbon black susceptibility measurements. Three different types of
CB were analyzed. The red curve corresponds to the CB779 which
was used in or study. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3-28 Simulation results for two spheres showing the field distribution in a
small region between the carbon black spheres. . . . . . . . . . . . . . 114
3-29 Simulation results for three spheres showing the field distribution in a
small region between the carbon black spheres. . . . . . . . . . . . . . 115
3-30 Simulation results for five spheres showing the field distribution in a
small region between the carbon black spheres. . . . . . . . . . . . . . 115
3-31 Schematic model showing the surface (red) and entangled (green) elastomer components in the vicinity of the CB particle. . . . . . . . . . 117
4-1 CT image of Sample A. The sample holder, seen as the black outline,
is 2 cm in diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4-2 CT image of Sample B. The sample holder, seen as the black outline,
is 2 cm in diameter.
. . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4-3 UPPER PANELS: MASS taken at 3T while the sample is spun at
8KHz. LOWER PANELS: respective static spectra. . . . . . . . . . . 129
4-4 UPPER PANELS: MASS taken at 12T while the sample is spun at
8KHz. Oil has, at least, three peaks — alkanes to the right of water
and aromatics to the left of water. LOWER PANELS: respective static
spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4-5 MASS-Inversion Recovery on Sample A and Sample B at νr =8KHz
These results give for Sample A: T1 (water)=2.1s, T1 (oil1)=0.51s, T1 (oil2)=0.51s
and for Sample B: T1 (water)=1.04s, T1 (oil1)=0.56s, T1 (oil2)=0.58s.
Note that T1 of water in Sample A is close to its bulk value, while in
Sample B it is approximately half its bulk value. . . . . . . . . . . . . 131
4-6 MASS-CPMG on Sample A and Sample B (νr =8KHz, τe =125µs). . . 132
4-7 T1 and T2 distributions for the water peak in both Sample A and Sample B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
15
4-8 T1 and T2 distributions for the oil1 peak in both Sample A and Sample
B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4-9 T1 and T2 distributions for the oil2 peak in both Sample A and Sample
B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
16
List of Tables
1.1
TOSS time sequences, in units of the rotor period. . . . . . . . . . . .
2.1
Sideband intensities of the integrated spectrum along diagonal line of
the 2D spectrum in the sample with water in a pack of beads. . . . .
2.2
39
60
Fitting results for different mixing times. The sample is the packed
50µm glass beads filled with 1% water and 99% D2 O. . . . . . . . . .
69
3.1
Sample properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3.2
Experimental results: intensity ratio encapsulated/surface . . . . . . 103
3.3
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.4
Estimations of mean diameters of CB particles derived from SEM experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.1
Quantitative X-Ray Diffraction Data Analysis (Weight Percent) . . . 127
4.2
Longitudinal Relaxation Times . . . . . . . . . . . . . . . . . . . . . 135
4.3
Chemical Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
17
18
Chapter 1
Molecular diffusion effects on
magic angle sample spinning
1.1
Amplitude modulation and relaxation due to
diffusion in NMR experiments with a rotating
sample
1.1.1
Summary
The simultaneous effects of diffusion and coherent averaging by Magic Angle Sample
Spinning (MASS) are investigated both theoretically and experimentally for spins
diffusing in a spatially varying susceptibility field. The diffusion introduces a periodic modulation and a relaxation of the magnetization amplitude. The attenuation
exponent for the nth echo, at an early time is given by D0 γ 2 g 2 πn/(4ωr3 ), where ωr is
the rotor frequency, D0 , the diffusion coefficient, γ, the gyromagnetic factor, and a
factor g 2 which is related to, but different from the volume average of the gradient
squared.1
1
The material presented in this section is published in Chem. Phys. Lett. 332, 344-350 (2000).
19
1.1.2
Background
Nuclear Magnetic Resonance methods (e.g. imaging, diffusion, relaxation) are powerful non-invasive probes to study structure and transport in various porous media, from
biological to geological [1, 2, 3, 4, 5, 6]. Variations in the local bulk magnetic susceptibility introduce spatially varying local fields. In an inhomogeneous magnetic field,
the spin packets at different locations precess at different Larmor frequencies leading
to defocussing and to an apparent destruction of the total transverse magnetization.
These local fields pose a fundamental complication for NMR studies of spatially heterogeneous samples [1, 2, 3, 4, 5, 6]. The detrimental effects of the susceptibility
variations are well documented and they lead to a decrease in spectral resolution,
non-linearities (distortions) in NMR imaging, and incorrect measurements of molecular diffusion and spin-spin relaxation rates. In general, the effects of inhomogeneity
are removed by coherent averaging of the Hamiltonian which is performed by rotation
either in the spin [7, 8, 9, 10] or in the physical space [11, 12, 13, 14, 15, 16, 17] or
both. Refocusing occurs to the extent that the spins have not changed their positions
in the inhomogeneous field.
Spin echoes are observed when the defocussing effects of the inhomogeneities
are reversed by manipulation in spin space such as by Hahn Echo (HE) or CarrPurcell-Meiboom-Gill (CPMG) sequences [7, 8, 9, 10] or by a coherent averaging of
the field via spinning the sample, such as Magic Angle Sample Spinning (MASS)
[11, 12, 13, 14, 15, 16, 17]. In CPMG, after the initial π/2 pulse, a series of π pulses
are applied periodically at an interval of 2τ and echoes are observed at times in
between the pulses. In the MASS experiments, the sample is spun around an axis
√
which is tilted at the “magic angle” Arctan( 2) with respect to the static magnetic field and the magnetization is observed after the initial π/2 pulse. WAHUHA
(Waugh/Huber/Haeberlen) sequence [10] is an example of a spin manipulation scheme
which aims at removing the dipolar interactions in solids by averaging the spin degrees of the Hamiltonian through a series of interaction frames which can be likened
to the real space averaging of MASS. The MASS has become a standard technique
20
for obtaining high resolution NMR spectra in solids [11, 12, 13]. Recently it has been
shown that MASS also removes the susceptibility broadening [4, 5] which then allows
one to identify complex fluids in situ.
The refocusing is incomplete when the spins move in the inhomogeneous field. The
movement encodes the terrain – i.e. the geometrical restriction as well as the lengthscales involved in the field inhomogeneities. In spin manipulation experiments, the
decay of echo amplitude is widely used to investigate the diffusive processes [1, 18].
This is extremely useful in characterizing the inhomogeneous magnetic fields which
carry the finger-print of the geometry [19, 20, 21] and functionality. For example,
the susceptibility inhomogeneities are tissue specific and are important for functional
imaging such as BOLD (blood oxygenation level dependent) contrast [20]. Another
example is the determination of multiple length scales in complex rocks via the decay
in the internal inhomogeneous field [21]. There are a large number of studies that use
spin manipulation programs to characterize geometry and internal field via diffusion
[1, 2, 3, 21, 22, 23, 24, 25]. In comparison, the effect of molecular diffusion on MASS
has received much less attention [15, 16, 26].
During MASS, a given spin packet of transverse magnetization retraces the same
trajectory during each rotor period and an echo forms when it returns to its original
position at the end of each rotor period [13]. The addition of random molecular
diffusive motion means that the trajectory becomes, to some extent, random and this
gives rise to decoherence and signal attenuation. The purpose of this work is to go
beyond the previous studies and evaluate quantitatively the effects of translational
diffusion in a MASS experiment. A quantitative understanding of diffusion in MASS
is important if MASS is to become a tool for quantifying the local structure and
dynamics of soft matter. Soft matter exhibits significant molecular diffusion and
flow, and is present throughout biological, chemical and physical systems. We believe
that MASS will become an important tool in the study of these systems. In this
section we analyze the rotational echo heights as a function of the underlying physical
parameters. The novel feature of MASS in fluids, which arises due to the diffusion of
the molecules, is that the amplitude of a spin packet depends on its position and time,
21
unlike for solids where the amplitude is constant, independent of space and time.
1.1.3
Theory
Here we compute the magnetization in a MASS experiment. First, consider the
magnetic field seen by a spin in a complex system. The external field B0 ẑL polarizes
the system and induces an additional field B(r). When the susceptibility contrast is
small, the magnetization is mainly dipolar [4, 5] (i.e. the contribution from higher
order multipoles are negligible) and the extra field B(r) is given by the superposition
of fields from all the induced dipoles. For simplicity, take the superposition as a sum
over dipoles of strength Mj which are located at rj = (rj , θj , φj ). Extending to an
integral is trivial [5, 19]. The field at a given point of observation r i = (ri , θi , φi ), in
the rotor frame, is
Bz (ri , θi , φi , t) =
2
X Mj X
j
3
rij
e−inωr t d20,n [β]Yn2 (θij )
(1.1)
n=−2
where rij = (rij , θij , φij ) is the vector connecting the points ri and rj . In the lab
frame, the field is proportional to Y02 (θij ). The Wigner rotation matrix d20,n [β]e−inωr t
transforms the field from the laboratory frame to the rotor-fixed frame [27]. The z
axis of the rotor frame is tilted at an angle β with respect to zL , which is fixed in
the laboratory frame. Now we can use an addition theorem to expand Yn2 (θij ) in a
series proportional to
P
m
2
2
D0,m
[θi , φi ]Dm,n
[θj , φj ] [27]. After a few simplifying steps,
the local field can be written in the standard form as [16]
Bz (r, ωr , t) =
2
X
[cj (r) cos(jωr t) + sj (r) sin(jωr t)],
(1.2)
j=1
where c1 , c2 , s1 , s2 are general functions of position.
We consider a system of non-interacting one half spins. The Hamiltonian corresponds to the Zeeman interaction of the spins with the inhomogeneous dipolar
field B(r) arising from susceptibility contrast Eq. (1.2), and can be written as
H =
P
l=0,2
Pm=l
m=−l (−1)
m
R̂l,−m T̂l,m where R̂ and T̂ are second rank tensor oper22
ators representing, respectively, the spatial and the spin dependencies of the coupling
[14].
Next we consider how the transverse spin magnetization M (r, t) = MxL (r, t) +
iMyL (r, t), after an initial π/2 pulse, evolves under the action of the magnetic field
above (Eq. (1.2)). The subscript L denotes the lab-frame.
First, consider the case in the absence of diffusion. In this case the results are well
known from solid-state NMR [11, 12, 13]
M (r, t) = e−i γ
Rt
0
B(r,t0 )dt0
M (r, 0) = e−i Φ(r,t) M (r, 0).
(1.3)
A factor of e(−iω0 −1/T2B )t has been divided out of M (r, t), where ω0 = γB0 is the
Larmor frequency, T2B the bulk decay rate. Here
2
cj
γ X
sj
[ sin(jωr t) − (cos(jωr t) − 1)].
Φ=
ωr j=1 j
j
(1.4)
Note that the phase is a simple periodic function of time with period τr = 2 π/ωr . In
the absence of diffusion, this gives M (r, t + n τr ) = M (r, t). For an initial uniform
magnetization the amplitude of M (r, t) is a constant independent of position and
time. Each spin packet, in the absence of diffusion (and spin-spin interaction), evolves
independent of each other, in the local magnetic field seen by it. We will see below
that the effect of diffusion changes this result. Diffusion will mix these spin packets
during the measurement period. This we take up next.
There are many different ways of incorporating the stochastic motion of molecules.
Here we add an anti-hermitian term to the Hamiltonian [28, 29] to incorporate molecular diffusion: H → H + i h̄ D0 ∇2 . The equation of motion simplifies for the present
case of an isolated, one-half spin to the Bloch-Torrey [30] equation (1.5) below
∂M (r, t)
= D0 ∇2 M (r, t) − iγB(r, t)M (r, t)
∂t
(1.5)
where D0 is the diffusion coefficient. The direct integration of the Bloch-Torrey
23
equation leads to
Rt
2
0
0
M (r, t) = T e 0 (D0 ∇ −iγB(r,t ))dt M (r, 0).
(1.6)
Here T is the time ordering operator. The initial condition is M (r, 0) = constant.
Surface relaxation can be added by a proper boundary condition for the normal
derivative ∂M/∂n at the surface, but for now we neglect restriction or relaxation
effects which are negligible for the time scales of observation here (see below). The
general analytical solution of Eq. (1.5) for an arbitrarily complex susceptibility field
is not possible, even in the case of no spinning. Below we outline how to obtain a
perturbative solution and extract a universal scaling form for the relaxation exponent
which can be applied to experiments.
The leading order short-time behavior of Eq. (1.6) can be obtained in several
different ways, of which the most direct is by using a cumulant expansion [31, 32].
We have discussed elsewhere [22] these methods as they apply to HE and CPMG
sequences. Using the fact that ∇2 B = ∇2 M (r, 0) = 0 for the quasi-static case, and
keeping terms up to second order in B, we find that the signal is proportional to (for
a uniform pick-up coil)
M (t) =
·e−D0 γ
2
R
Rt
0
d3 rM (r, t) =
dt0 (∇
R t0
0
R
d3 r
B(r,t00 )dt00 )2 −
e
Rt
0
iγB(r,t0 )dt0
(1.7)
M (r, 0).
After inserting Eq. (1.2) in Eq. (1.7) and carrying out the time integrations, we
see that the exponent has several periodic terms of frequencies ωr , 2ωr , 3ωr and 4ωr
and a non-periodic term giving attenuation:
M (t, ωr 6= 0) = M (0)
R
3
d re
− 81
D0 t γ 2 g 2
2
ωr
e−pR (ωr t)−iΦ(ωr t)
(1.8)
g 2 = 4(∇c1 )2 + (∇c2 )2 + (∇s1 )2 + 3(∇s2 )2 + 8∇s1 · ∇s2 .
24
The real part of the periodic piece of the exponent is
pR =
4
D0 γ 2 X
[fj (cos(jωr t) − 1) + hj sin(jωr t)].
8 ωr 3 j=1
(1.9)
Here the functions fj and hj involve gradients, for example f4 = 12 ∇c2 · ∇s2 .
As noted before, c1 , c2 , s1 , s2 are functions of position. For the special case, where
the gradients of c1 , c2 , s1 , s2 are constant vectors, Eq. (1.8) becomes exact for unrestricted diffusion. In the latter case, Torrey’s [30] method applies. The solution is
exact for D0 = 0. The real part pR is proportional to D0 /ωr3 and has information of
length-scale via its dependence on the derivatives such as ∇c1 etc. The phase part, Φ,
is independent of D0 and depends on c1 etc., but not on their spatial derivatives, and
Φ falls off as 1/ωr . These results can be appreciated without a recourse to the formal
apparatus of Kubo et al. [31] by extending Torrey’s [30] method to a time-dependent
field gradient g(t) which is, otherwise, a constant in space. This gives a time dependent wave vector k(t) =
Rt
0
dt0 g(t0 ). It may be checked by substitution that the
solution of Bloch-Torrey Equation is given by M (t) = M (0)e−D0
Rt
0
dt0 k(t0 )2
× e−ik(t)·r .
Thus, for a periodic g(t) the amplitude term is also modulated periodically.
We see that there is a rather fundamental difference between the MASS spectra
of solids and of fluids: the amplitude part for fluids is non-trivial. In this work we
concentrate on the attenuation term. The periodic terms give rise to the sidebands,
and we are in the process of analyzing them. At echo times t = 2nτ = n τr = n 2π/ωr
the periodic terms associated with ∇B return to zero, and phase factors return to zero.
Thus, we can extract the overall decay by stroboscopically measuring the transverse
magnetization at times t = n τr , i.e., the heights of the rotary echoes. We see that the
attenuation exponent for the nth echo, at (early) times t, is given by D0 γ 2 g 2 πn/4ωr3 .
When we compare with CPMG, if τ is the time between the initial π/2 pulse and
the first π pulse of the CPMG pulse sequence, the 2τ time between two consecutive
CPMG echoes corresponds to 2π/ωr , the time interval between the MASS echoes.
Next we examine how this result behaves for extremely slow rotational frequencies.
This will give a result for static samples which we will extend to CPMG on static
25
samples. If τ is the period between an echo and the π pulse in CPMG, the echo-period
for CPMG is 2τ , while the echo-period for MASS is τr = 2π/ωr . When ωr t << 1, a
part of pR cancels the D0 γ 2 g 2 t/(8ωr2 ) term, and the FID (FID: free induction decay)
exponent, with a leading correction of order ωr , becomes (D0 γ 2 )(1/3)[(∇c1 +∇c2 )2 t3 +
(1/4)(∇c1 + ∇c2 ) · (∇s1 + 2∇s2 )t4 ωr + ...]. For a non rotating sample with ωr = 0 this
FID result can be used to derive the corresponding relaxation in CPMG: [7, 8, 9, 22]
M (2nτ, ωr = 0) = M (0)
Z
2
d3 r e− 3 D0 nτ
g12 = (∇c1 + ∇c2 )2 .
3
γ 2 g12
(1.10)
For CPMG, the Hamiltonian is effectively periodic but non-sinusoidal, and it
takes up only one magnitude, but with two different signs, one before and the other
after each π pulse. The Hamiltonian is a product of a space part, R̂, and a spin
part, T̂ . MASS manipulates R̂, while, CPMG/HE experiment [7, 8, 9] controls T̂ .
The MASS experiments can be thought of as being complementary to the CPMG
experiments. From dimensional analysis alone, we expect the decay exponents to be
similar for MASS and CPMG. The actual numerical coefficients are different, and this
requires the detailed analysis given here. It will be interesting to use this difference
to better characterize local gradients (see below). The main difference, to reiterate,
is a frequency modulation in the MASS amplitude which is not present in the CPMG
amplitude.
1.1.4
Experimental results and discussions
The predicted scaling behavior for CPMG and MASS decay exponents is clearly seen
in the experimental data shown in Fig. 1-1. All the spectra were taken at 200 C and
at 500 MHz using a Bruker DRX500 spectrometer. MASS data were taken using
a high resolution 1 H/13 C MASS probe. Both the MASS and the CPMG data were
taken on the same sample – 50 µm glass beads packed in a 4mm MASS Zirconia
rotor filled with water. This sample has a relatively well defined geometry and is a
representative model system for heterogeneous, porous materials. Figure 1-1 shows
26
50 µm Glass Spheres
0
τr = 2τ = 0.400 ms
0.500 ms
-0.2
0.625 ms
0.714 ms
ln(A) - ln(A 0)
-0.4
-0.6
-0.8
CPMG
-1
MASS
-1.2
0
0.2
0.4
0.6
0.8
1
n/ω3r
1.2
( sec / rad )3
1.4
1.6
1.8
2
-11
x 10
Figure 1-1: Comparison of CPMG (static sample) and MASS (rotating sample) attenuation data on a pack of beads as a function of ωn3 (n is the echo number, ωr = 2π/τr
r
the rotor frequency. For purposes of comparison, we express the static CPMG data
in using 2τ = τr . Note that τr is the time between two echoes for MASS and 2τ
is the time between two echoes for CPMG). The early times data show the “free
10 n
and
diffusion” behavior. The solid lines correspond to ln(A/A0 )M ASS = − 14.9×10
ω3
r
10 n
.
− 7.5×10
ωr3
At later times the effects of restriction and the breakln(A/A0 )CP M G =
down of other approximations are apparent.
both CPMG and MASS data.
The relaxation time of water in beads is T2 ≈ 30ms and the diffusion length at
a rotor frequency of 1400Hz is about 1.2 µm, which is much smaller than the bead
size. The rotor period for MASS and the corresponding pulse spacing for CPMG were
chosen to be in the same range such that the effect of restriction is minimized and
the results can be compared and contrasted. The early time CPMG and MASS data
can be fitted respectively by: ln(A/A0 )M ASS = − 14.9×10
ωr 3
10
− 7.5×10
ωr 3
n
10
n
and ln(A/A0 )CP M G =
, where ωr is in radians per second. The lines in Fig. 1-1 are obtained by a
simultaneous fit of all the CPMG and all the MASS data respectively. The theoretical
results above are for unrestricted diffusion and naturally hold only for the early times.
27
The early time behavior demonstrates n/ωr3 behavior for the decay exponent. The
effects of restriction are seen clearly by the decay becoming less than the one given by
the free diffusion result. For short times, < g12 >= (∇B(r))2 controls the CPMG/HE
experiment [7, 8, 9, 22]. g1 is a characteristic size of the gradient of B(r), i.e. g1 =|
∇B |rms is the square root of the volume average (∇B(r))2 of the square of the
gradient. A similar factor g 2 applies to the MASS attenuation. Naturally, as noted
above, the difference between the experimentally measurable quantities, g 2 and g12
carries information about the geometry. Higher order spatial derivatives become
important at later times, as do the effects of restriction. By an analysis of the BlochTorrey equation, we can identify three main regimes of decay for CPMG: motionally
averaged, localization, and short time [22]. A similar analysis will apply to MASS.
The shortest length scale determines the asymptotic regime. In the short-time regime
√
the diffusion length LD = D0 τ is smaller than any other characteristic length of
the problem. The complete analysis of the effects of restricted molecular diffusion
[23, 24, 25] will be an important step in understanding compartments in biological
and geological samples.
A quantitative preliminary analysis of the decay in total magnetization under
MASS is given below. Assuming that the susceptibilities are isotropic, each glass
bead, of radius Rj , gives rise to a dipole which is located at its center and whose
strength Mj is given by Mj = [(µglass − µwater )/(µglass + 2µwater )]Rj3 B0 . The total
field is a superposition of these dipolar fields and may be computed numerically quite
easily [19]. However, as the dipolar field drops off rather rapidly with distance r,
the nearest single sphere may account for most of the dipolar field experienced by a
molecule diffusing over a short period of time. In the next paragraph we estimate
the decay exponent by considering only a single sphere to approximate the magnetic
field.
A second second-cumulant expansion is used to evaluate the integration over the
sample volume in (1.8). By performing first the angular average, the second cumulant,
at a distance r, due to a single dipole of moment m is D0 γ 2 m2 (18 ωr t −16 sin(ωr t)−
sin(2 ωr t))/(r 8 ωr3 ). Before integrating over r, it is instructive to note that the MASS
28
decay exponent for a single sphere at a distance r is 18D0 tγ 2 m2 /(r8 ωr2 ). The decay
exponent for the n − th CPMG echo at the time 2nτ is 8D0 τ 3 nγ 2 m2 /r8 . Since for
a single sphere, the ratio of the attenuation factors is
ln(A/A0 )M AS
ln(A/A0 )CP M G
=
9
,
(2π 2 )
using a
porosity of φ = 0.3, and ωL = 2π500 ×106 rad/sec, D0 = 2 ×10−5 cm2 /sec, R = 50µm,
∆χ = 10−6 [SI] and integrating over a thin shell of half the radius gives an attenuation
coefficient for MASS 3 × 1010 /(nωr3 ), which differs by a factor of 5 from the observed
data. This difference can be easily due to a number of factors, such as the fact that
we use only a single sphere to estimate the local magnetic field, we use the second
cumulant approximation twice, we assume that ∆χ is isotropic and uniform, and we
use only a rough estimate of ∆χ. We are in the process of performing a more detailed
study aimed at overcoming these limitations.
1.1.5
Conclusions
MASS is the premier technique in high resolution solid state NMR for simultaneous
observation of different species with different isotropic chemical shifts. For soft matter, MASS has a similar potential, but the effects of diffusion have to be taken into
account and exploited. For example, the relaxation of individual components in a
complex system can be studied by combining MASS with CPMG/ inversion recovery
[26]. In solids, combined MASS and spin manipulation techniques [13] are commonly
exploited to extract important information about the individual species. An extension
of these methods for soft matter will require an accounting for the relaxation and amplitude modulation due to diffusion, as described above. In the absence of diffusion,
the NMR signal amplitude under MASS is independent of position and time, and the
phase is a periodic function of the rotor frequency ωr . This frequency modulation
reports on the range of susceptibility fields. In fluids, the amplitude of a spin packet
has modulations and attenuations which carry information on the molecular dynamics
and the landscape (susceptibility and restrictions). MASS provides a convenient and
powerful view of the structure and dynamics of soft matter. This study provides the
foundation for future analysis of restriction effects [23, 24, 25]. The method outlined
here will also be useful for analysis of sidebands in the frequency domain spectrum,
29
as well as in designing experiments with combined spin manipulation and MASS [13]
which are aimed at suppression and manipulation of sidebands.
30
1.2
Manipulation of phase and amplitude modulation of spin magnetization in magic angle
spinning NMR in the presence of molecular
diffusion
Nuclear Magnetic Resonance experiments with a spinning sample (Magic Angle Sample Spinning (MASS)) are used to remove the line broadening in composite systems,
where the susceptibility contrast of its constituents gives rise to an inhomogeneous
field that causes a line broadening and obscures chemical information. The NMR
signal in these experiments has a phase and an amplitude part. In the absence of
diffusion, i.e. in the MASS spectra of solids, the amplitude of the signal from an
isochromat is a constant independent of position and time and the phase is a periodic function of the rotor frequency νr . In fluids, the amplitude of a spin packet is a
function of its position and time. The amplitude modulation and relaxation in diffusive MASS encodes the dynamics of motion and the landscape (geometry of pores
and field gradients) probed by the motion. Here we use spin manipulation – total
suppression of sidebands (TOSS) to suppress the effects of phase with the goal of
isolating the amplitude term. By the TOSS sequence the phase factor at time t for
a spin packet at an azimuthal angle φ is made to depend on φ only as a function of
ωr t − φ, which suppresses the sidebands in solids upon an integration over φ. Due
to molecular diffusion, the amplitude part depends on φ, and, thus, diffusive TOSS
cannot suppress the sidebands. The residual sidebands carry the information of dynamics and pore and magnetic field geometry, in addition, by reducing the size of the
sidebands, TOSS is of course, also useful in identifying various fluid components in
situ. The diffusive MASS gives a measure of the spread in local fields and diffusive
TOSS gives a measure of the spread in local gradients.2
2
The material presented in this section is published in J. Chem. Phys.
(2001).
31
114( 13) 5729-5734
1.2.1
Partial suppression of sidebands with TOSS
The purpose of this work is to investigate the effects of diffusion on high resolution
NMR spectra of fluids in complex and porous media obtained in a spinning sample
with simultaneous spin manipulation by a sequence of radio frequency (RF) pulses.
The goal of these experiments is to obtain, in addition to the chemical analysis of
the fluids in situ, information about the dynamics of the molecular motion and the
underlying landscape, i.e. the restricted geometry and field inhomogeneity.
High resolution NMR spectroscopy has become a standard tool for identifying
chemical constituents and their structure for simple liquids and solids. However, in
complex and composite media which are encountered both in natural and man made
systems, high resolution NMR spectroscopy has not realized its full potential due to
the presence of the inhomogeneous magnetic field which arises out of the susceptibility
differences of the constituents. The high fields required to separate chemical shifts
make the problem worse as the spread of the inhomogeneous magnetic field, i.e. the
line width, grows with the applied field.
In solids, the interactions among nuclei produce a broad line that obscures the
chemical information. Generally in NMR experiments this broadening is removed via
a coherent averaging of the Hamiltonian which is performed by a rotation either in
the spin or in the physical space or both: the Hamiltonian being a product of spatial
tensor with spin operators [13, 16, 15, 33, 34]. In the MASS experiments, the sample
√
is spun around an axis which is tilted at the “magic angle” β = Arctan( 2) with
respect to the static magnetic field. MASS is the premier technique for obtaining
high resolution NMR spectra in solids [13, 11, 12].
The techniques that are widely used in the solid state NMR to control the broadening of NMR signal are expected to be the natural tools for dealing with the line
broadening in complex systems [35, 4, 5].
A rather fundamental difference between the MASS spectra of solids and of fluids
is that the amplitude of a spin packet for fluids depends on position and time while for
solids the amplitude remains a constant, independent of space and time. As a result,
32
the amplitude of total observed signal in fluids is time dependent: it is a function
which decays with time multiplied by a part which does not decay but is a periodic
function of ωr t. The modulus of the amplitude for liquids has information on the
length-scales via its dependence on the derivatives of the field and on the fluid type
via its dependence on diffusion coefficient. Both in solids and liquids, the phase part
is independent of diffusion, but depends on the field distribution but not on their
spatial derivatives.
The relaxation term in magnetization in the MASS experiments with fluids is
interesting and is similar to the well known relaxation of magnetization by diffusion
as seen in echoes of non-spinning samples obtained using spin manipulation experiments such as Hahn echo, CPMG and stimulated echo [7, 8, 9, 1, 18, 21]. The latter
is widely used to investigate the diffusive processes and to characterize the inhomogeneous magnetic fields which carry the finger-print of the geometry [21, 37, 20] and
functionality. In comparison, the effect of molecular diffusion on MASS has received
much less attention [16, 15, 26].
Returning to the novel feature of MASS in fluids, i.e. the periodic modulation
of amplitude, it affects the phase and the amplitude of the sidebands just like the
periodic modulations in the phase. In fact, if the modulations in the phase term
were not present, sidebands would be purely due to amplitude modulations. As the
time dependence of the amplitude term is purely due to motion, it would be useful
to separate out the amplitude modulation part and study the information coded by
motion. Here the spin manipulations are introduced with the aim of suppressing the
phase terms, which are independent of D0 .
To suppress the interference of the phase part, one may use one of the numerous
spin manipulation schemes that are designed to manipulate the phase part and are
available in the literature [16, 15]. An example is the TOSS [33] sequence, which was
designed to suppress the sidebands in a MASS experiment.
While the sidebands are suppressed by TOSS in solids, they are only partly suppressed in liquids. By application of TOSS the phase factor at time t for a spin packet
at an azimuthal angle φ is made to depend on φ only as a function of ωr t − φ, which
33
suppresses the sidebands in powdered solids, upon an integration over φ. Here we
find that due to molecular diffusion, the amplitude part in TOSS depends on φ, in
addition to on ωr t − φ, and, thus, the TOSS cannot suppress the sidebands.
The fact that non-suppression of sidebands under TOSS being due to motion
can be exploited to study the effects of molecular diffusion and the length scales of
magnetic inhomogeneity and geometrical restriction. In a wide range of areas, such
as oil recovery, heterogeneous catalysis and biological perfusion, the diffusion of fluids
in porous media plays a vital role.
1.2.2
Equation of motion of magnetization for a spinning
sample in the presence of diffusion
In our problem, the Hamiltonian H corresponds to the interaction of the spin with
the inhomogeneous dipolar field B(r) arising from susceptibility contrast [4, 5, 37].
To be more specific, B(r) is the local field in the zL -direction obtained from the superposition of fields from all the dipoles that are induced by the susceptibility contrast
and have the dipole moment in the zL -direction proportional to the applied field B0 ẑL
times the susceptibility contrast [4, 5, 37]. The dipolar interaction, when transformed
to the lab frame, gives rise to a field which is time dependent at frequencies ωr and
2ωr [11]. The position of a spin packet is defined with respect to the sample-frame
which is rotating. The local field can be written in a compact form as:
Bz (r, ωr , t) = c1 (r, θ) cos(ωr t − φ) + c2 (r, θ) cos(2 ωr t − 2φ)
(1.11)
In general, c1 , c2 are arbitrary functions of radial and polar coordinates, (r, θ).
As explained earlier, to incorporate molecular diffusion, we add an anti-hermitian
term to the Hamiltonian. For the present case of an isolated, one-half spin, the
Bloch-Torrey [30] equation is Eq. 1.5. The magnetization density transverse to the
applied field B0 ẑL is M (r, t) = MxL (r, t) + iMyL (r, t). The components in the labframe are denoted by xL , yL , zL . A factor of exp(−iω0 − 1/T2B )t has been divided out
of M (r, t), where ω0 = γB0 is the Larmor frequency, 1/T2B the bulk decay rate. A
34
direct integration of the Bloch-Torrey equation leads to Eq. 1.6.
For an initial, spatially uniform magnetization, we can use the fact that ∇ 2 M (r, t =
0) = 0, and for the quasi-static case ∇2 B(r, t) = 0 and keep terms up to second order
in B. We find that the signal is proportional t0 (for a uniform pick-up coil)
M (r, t) = M (t0 )e−pR (r,t,t0 ) − i Φ(r,t,t0 )
Φ(r, t, t0 ) = γ
Rt
0
0
t0 B (r, t ) dt
(1.12)
The real part, pR (r, t, t0 ) is a time and space independent constant for solids and
becomes non-trivial with diffusion. In the diffusive case, pR (r, t, t0 ) has a non-periodic
decay part proportional to D0 t/ωr2 and terms which are periodic in ωr t. At echo times
t = n 2τ = n 2π/ωr the periodic terms associated with ∇B return to zero, the phase
factors return to the same value, thus we can extract the overall decay by measuring
the heights of the rotary echoes. We see that the attenuation exponent for the n − th
echo, at (early) time t, scales as D0 γ 2 g 2 nt/ωr2 .
1.2.3
MASS and TOSS without diffusion
Before discussing the effects of diffusion on TOSS, it is useful to recall a few relevant
facts for the case without diffusion.
To recall TOSS [13, 33], consider the phase part in Eq. (1.12) of a spin packet
whose azimuthal coordinate is φ:
e− i Φ(ωr t) = f (φ − ωr t) f ∗ (φ)
f (φ − ωr t) = e−iγ
Rt
B(r,θ,φ,t)dt
(1.13)
(1.14)
Here we follow the notation used in the reference [16]. TOSS is designed to remove the
f ∗ (φ) term form equation (1.13), so the φ variable appears only as ωr t − φ, such that,
in absence of diffusion terms, an integration over φ allows only the center-band to
survive. A nice discussion of this is given in the book by Schmidt-Rohr and Spiess [16].
The simplest TOSS sequence [33] consists of four properly placed π pulses applied at
35
900
1800
1800
1800
1800
t0 = 0
t1
t2
t3
t4
acq
t5
1
u(t)
t
Figure 1-2: The TOSS sequences that we used consists of four π pulses followed by a
waiting time, all chosen to satisfy TOSS conditions.
given times t1 , t2 , t3 , t4 followed by a waiting time before the start of acquisition. The
effects of π pulses can be incorporated, most conveniently, by multiplying B(r, t) by
a function of unit magnitude u(t) which simply changes its sign upon an application
of a π pulse [22] B(r, t) → u(t) B(r, t). See Fig. 1-2.
Thus, the phase at a given time t after the fourth pulse is
Φ(r, t, 0) = Φ(r, t, t4 ) − Φ(r, t4 , t3 ) + Φ(r, t3 , t2 ) − Φ(r, t2 , t1 ) + Φ(r, t1 , 0)(1.15)
t5 denotes the start of data acquisition and is chosen to satisfy the Hahn spin echo
condition:
2 (−t1 + t2 − t3 + t4 ) = t5
(1.16)
The TOSS times are chosen [33] so as to fulfill,
2 [cos(ωr t1 ) − cos(ωr t2 ) + cos(ωr t3 ) − cos(ωr t4 )] = 1
sin(ωr t1 ) − sin(ωr t2 ) + sin(ωr t3 ) − sin(ωr t4 ) = 0
(1.17)
With these conditions, we see that the phase at the beginning of the acquisition
depends only on ωr t − φ and f ∗ (φ) disappears from Eq. (1.13). See Ref.[16] for more
details.
36
1.2.4
MASS and TOSS with diffusion
The main effect of diffusion is coded in the amplitude part of magnetization [35] in
the MASS spectra of liquids, and thus by isolating the amplitude from the phase
part we should obtain additional information encoded via diffusion. In this section
we consider two simple cases which explicitly demonstrate why the sidebands are
not suppressed in the presence of diffusion. A general solution is not possible but
we will illustrate this by two simple cases: (i) diffusion in the presence of simple
model phase factor; (ii) diffusion in the presence of a single dipole. We will show this
by computing the real part of the signal amplitude. To compute the magnetization
after the waiting time requires a solution of (1.5) with an initial condition that the
magnetization is non-uniform in space due to a position dependent phase term. There
is some attenuation during the total period from time t = 0 to t5 , and the attenuation
of the spin packet will depend on position, but we will neglect this as its effect is
much smaller than that of space-dependent phase modulation of magnetization. Also
incorporating additional position dependent attenuation terms is rather cumbersome
and only obscures the physics.
1.2.5
Model phase factor
Let us consider a spin packet at (r, θ, φ) and assume that at a time t5 the transverse
magnetization has acquired a phase, which we assume to be of the following simple
form:
M (r, t5 ) = e−iΦ(r,t5 ) M (0, t5 ),
γb
Φ(r, t) =
cos(ωr t − φ)
ωr
(1.18)
where M (0, t5 ) and b, the local magnetic field, are assumed to be independent of
spatial coordinates.
37
The phase term of a spin packet during the acquisition is:
Φ(r, t, t5 ) =
γb
ωr
[cos(φ − ωr t5 ) + (cos(φ − ωr t) − cos(φ − ωr t5 ))]
(1.19)
The first term in right hand side of the equation above is the initial phase at time
t = t5 and the second term is the additional phase accumulated during acquisition.
The amplitude is computed by taking the time integral of (5Φ)2 as given by the
lowest order perturbation theory. Then, by defining the new time variable such that
t5 = 0 we find that the magnetization during acquisition is:
M (r, t) = M (t0 )e−pR (r,t,t0 ) − i Φ(r,t,t0 )
Φ(r, t, t0 ) =
pR =
γb
ωr
γ 2 b2 D 0
2 ωr2 r 2 sin2 θ
cos(φ − ωr t)
³
t−
sin(2 φ)−sin(2 φ−2 ωr t)
2 ωr
(1.20)
´
This equation shows that in addition to usual time dependent terms which combine
with φ in the form of ωr t − φ, a factor of sin(2 φ) appears in the exponent. Therefore,
integration over the variable φ no longer removes the time dependence. Thus we have
non-zero side-bands.
It can also be seen that different TOSS sequences will produce different phase
parts, the total time being different, and therefore the concomitant real parts will
have different φ dependencies leading to different forms of sidebands from one TOSS
sequence to another.
1.2.6
Diffusion in the presence of a single dipole
Next, consider diffusion in the presence of a single magnetic dipole. In the experiments
with a pack of beads, the field on a proton due to its nearest bead is maximum, as
the field falls off as the distance cubed. So a solution of our problem in the presence
of a single dipole is an excellent approximation, at least for the short-time regime.
The field produced by a dipole of strength M in the rotor-frame is:
´
√
M ³
2
2
cos(φ
−
ω
t)
sin(2
θ)
cos(2
φ
−
2
ω
t)
sin
θ
+
r
r
r3
38
(1.21)
Table 1.1: TOSS time sequences, in units of the rotor period.
1
2
t1
0.188821
0.122789
t2
0.769947
0.467027
t3
0.811179
1.423923
t4
1.230053
2.200243
t5/acq
2.00000
2.24111
Using arguments similar to those given above, we find
√
γ 2 M 2 D0
[
1773
ω
t
−
465
ω
t
cos(4
θ)
+
625
2 sin(2 θ) sin(φ)
r
r
192 r8 ωr 3
√
√
− 120 2 sin(4 θ) sin(φ) − 1344 sin2 (θ) sin(2 φ) − 320 2 cos(θ) sin3 (θ) sin(3 φ)
pR =
− 30 sin4 (θ) sin(4 φ) + 12 cos(2 θ) (19 ωr t + 80 sin2 (θ) (− sin(2 φ)
√
+ sin(2 φ − 2 ωr t))) − 624 2 sin(2 θ) sin(φ − ωr t)
√
+ 120 2 sin(4 θ) sin(φ − ωr t) + 1344 sin2 (θ) sin(2 (φ − ωr t))
√
+ 320 2 cos(θ) sin3 (θ) sin(3 (φ − ωr t)) + 30 sin4 (θ) sin(4 (φ − ωr t)) ]
(1.22)
Again, because of the factors sin(2 φ), sin(3 φ)... in the exponent, an integration
over the variable φ no longer removes the time dependence and the side-bands have
finite intensity.
1.2.7
Experimental results
In this section we study the effect of different TOSS sequences for water in a pack of
beads and show that the sidebands are quite different, as argued above.
There is a whole class of solutions to these equations [34]. The time sequences for
TOSS [38] used in the experiments below are tabulated in Table 1.1.
Among the sequences in Table 1.1, the second one is good as the intervals between
the pulses are more evenly distributed. But the total time needed to implement this
sequence is a little longer than that for the first one and the relaxation effects can be
39
greater. In our experiment, we mostly use the first sequence.
All the spectra were taken at 200 C and at 500 MHz using a Bruker DRX500
spectrometer equipped with a high resolution 1 H/13 C MASS probe with gradient
coils. Both the MASS and the TOSS data were taken on the same sample – 50 µm
glass beads packed in a 4mm MASS Zirconia rotor filled with water. A static proton
spectrum of water in the pack of glass beads shows a single broad water resonance,
of width of about 2kHz [35]. The relaxation time of water in beads is T2 ≈ 30ms and
the diffusion length at a rotor frequency of 1400Hz is about 1.2 µm, which is much
smaller than the bead size. Figure 1-3 shows both TOSS and MASS data.
It is instructive to compare the results on water to that of a solid sample. In
order to get the static linewidth in a solid similar to that of water in 50 µm glass
beads, we used a wax – poly( ethylene glycol) sample. This sample has a static
spectrum similar to that of water in a pack of 50 µm glass beads. The wax spectrum
has three chemically distinct resonances. In Fig. 1-3, the MASS of wax at 1kHz
shows two center bands near the origin, and a third small center peak around 4ppm
which becomes more visible under TOSS. We see that, even for a relatively slow
rotational frequency, a TOSS sequence removes the sidebands quite well for wax,
whereas for water the side bands remain quite prominent. A slow rotational frequency
was deliberately chosen to emphasize the amplitude terms.
A detailed analysis of the side-bands is beyond the scope of the present chapter,
however, we can make a few simple observations based on the general features of the
phase and amplitude modulations. The main behavior of the signal comes from the
phase term, since it causes the spins to fan out over the sample and the total signal,
integrated over the sample volume, becomes very small or even zero except when an
echo forms. The amplitude modulations, on the contrary, are purely real and do not
cause a destructive interference. The next point to note is that the phase exponent,
i (∆Ω/ωr ) cos[m(ωr t − φ)], m = 1, 2, a periodic function of time, has a relatively
large amplitude. The cosines are bounded between ±1, so that the width of the phase
variation is given by the line-width ∆Ω/ωr relative to the rotor frequency. The number
of side-bands N observed in MASS is of the order N ∼ ∆Ω/ωr , i. e. for a given
40
value of ∆Ω/ωr , the side-bands with an order greater than ∆Ω/ωr have negligible
amplitude. The width of phase modulations, ∆Ω/ωr , generally, is much larger than
the corresponding width of the amplitude term (∆Ω/ωr )2 (LD /Lc )2 , where Lc is a
characteristic length for the variation of the local field, i.e. the characteristic local
field gradient is ∆Ω/Lc γ and LD =
q
D0 /ωr is a diffusion length of the order of the
distance that the molecules diffuse during a rotor period. In addition to the periodic
modulation, the amplitude decays due to diffusion. The side-bands are broadened by
the diffusive relaxation.
In summary, when diffusion is relatively small (LD /Lc ) << 1, the amplitude
modulation can be treated as a perturbation to the phase modulation. Thus, the
number of side-bands in MASS with diffusion is about the same as for the case
with negligible diffusion, and with TOSS, we expect that the side-bands are almost
suppressed. From the structure of the amplitude term we expect that only up to
about four additional side-bands will have any significant strength when diffusion is
present. In fact, the side-bands up to N ∼ (∆Ω/ωr )2 (LD /Lc )2 , will have a significant
strength in TOSS.
To quantify these ideas, consider the root mean square frequency spread ∆ν (2) =
qR
dνν 2 S(ν), where S(ν) is the frequency domain signal, which is normalized such
that
R
dνS(ν) = 1. For MASS this gives a measure of the frequency spread in the
local fields and for the TOSS gives a measure of the spread in the local gradients, as
noted above. We find that
(2)
(2)
∆νwater,M ASS ≈ 2080Hz, ∆νwax,M ASS ≈ 1586Hz,
which are about the same. The corresponding spreads in TOSS:
(2)
(2)
∆νwater,T OSS ≈ 541Hertz, ∆νwax,T OSS ≈ 117Hz,
differ by about a factor of five. It may be noted, in passing, that the particular wax
used here has a few mobile components which give rise to the vestiges of rather small
side-bands under TOSS.
41
Wax in glassbeads
Water in glassbeads
νr = 1kHz
MASS
x4
x4
TOSS
10
0
-10
Frequency (kHz)
10
0
-10
Frequency (kHz)
Figure 1-3: Comparison of wax and water spectra, both embedded in a pack of beads
and subjected to the first TOSS sequence from Table 1.1.
The different TOSS sequences will give different spectra (see Fig. 1-4). The
sidebands are greatly suppressed, but the amplitudes and phases of sidebands are
different with different sequences. Two illustrations are shown in Fig. 1-4.
1.2.8
Conclusions
We used TOSS to study fluids in a porous medium with the goal of showing that
there are unique differences due to molecular diffusion. These differences obviously
carry the information about the dynamics of the molecular motion and the landscape,
i.e. the restricted geometry and field inhomogeneity, over which this motion is taking
place.
Sidebands are not fully suppressed by TOSS in the presence of diffusion and we
explain this by using two simple models. TOSS makes the phase term of the complex
magnetization depend on the azimuthal angle φ through φ − ωr t. In the absence of
42
x 2.5
8
0
Frequency (kHz)
Figure 1-4: Comparison of the two different TOSS sequences for water in a pack of
glass beads (νr =1kHz). The thin line corresponds to the first TOSS time sequence
from Table 1.1 while the thick line corresponds to the second sequence.
43
diffusion, the amplitude of the magnetization is independent of position and time so
that only the center-band survives. This is because an integration over φ removes
the time dependence via a change of variable φ → φ − ωr t. However, in liquids, the
amplitude term has additional φ dependence and this transformation does not kill
the sidebands. The r.m.s. frequency spread for MASS is about the same for both
wax and water, but for TOSS the r.m.s. frequency spread for water is about five
times greater than that for wax. For MASS the r.m.s. spread gives a measure of the
frequency spread in the local fields, and for TOSS it gives a measure of the spread in
the local gradients times a diffusion length.
MASS is the premier technique in high resolution solid state NMR. The frequency
modulation of a spin packet reports on the range of susceptibility fields. In fluids, the
amplitude modulations and attenuations carry information of the landscape (susceptibility and restrictions). Thus, for soft matter, diffusive MASS has even a greater
potential since the molecular motion reports on the correlation lengths for susceptibility gradients which are related to structure [21]. Diffusive TOSS provides a convenient
and powerful method for focusing on the amplitude modulation and the additional
information contained there in. This work sets up the physics of probing the dynamics
from residual MASS sidebands and shows that information about important length
scales, the characteristic length for variation of local gradients and two characteristics
frequencies (rms frequency spread, line-width) can be extracted from the data. This
information is not otherwise available. However, to address the inverse problem will
require a broader set of measurements and new methods of analysis that need to be
developed.
44
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(1991).
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John Wiley, New York (1996).
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385 (1998).
[6] W.E. Maas, F.H. Lukien, D. G. Cory, J. Am. Chem. Soc., 118 (15), 13085
(1996).
[7] E.L. Hahn, Phys. Rev., 80, 580 (1950).
[8] H.Y. Carr, E.M. Purcell, Phys. Rev., 94, 630 (1954).
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[12] I.J. Lowe, Phys. Rev. Lett., 2, 285 (1959).
45
[13] R.G. Griffin, Rotating Solids, Encyclopedia of Nuclear Magnetic Resonance,
D.M. Grant, R.K. Harris (Ed.s); John Wiley, New York (1996).
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[15] L. Elmsley, A. Pines, Lectures on Pulsed NMR, second edition, Proc. Fermi
School, Soc. Italiana di Fisica, CXXIII, (1994).
[16] K. Schmidt-Rohr, H.W. Spiess, Multidimensional Solid-state NMR and Polymers, Academic Press, New York (1994).
[17] G.A. Williams, H.S. Gutowsky, Physical Rev., 104, 278 (1956).
[18] E.O. Stejskal, J.E. Tanner, J. Chem. Phys., 42, 288 (1965).
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[20] J.W. Belliveau, D.N. Kennedy, R.C. McKinstry, B.R. Buchbinder, R.M. Weisskoff, M.S. Cohen, J.M. Vevea, T.J. Brady, B.R. Rosen, Science, 254, 716 (1991).
[21] Y-Q Song, S. Ryu, P.N. Sen, Nature, 407(6792), 178 (2000).
[22] P.N. Sen, A. André, S. Axelrod, J. Chem. Phys., 111, 6548 (1999).
[23] T. De Swiet, P.N. Sen, J. Chem. Phys., 100, 5597 (1994).
[24] T. De Swiet, P.N. Sen, J. Chem. Phys., 104, 206 (1996).
[25] P.P. Mitra, P.N. Sen, L.M. Schwartz, P. LeDoussal, Phys. Rev. Lett., 65, 3555
(1992) and references therein.
[26] See, for example, references cited in P. Weybright, K. Millis, N. Campbell, D.G.
Cory, S. Singer, Magn. Reson. Med., 39, 337 (1998).
[27] M.E. Rose, Elementary Theory of Angular Momentum, Wile, New York (1957).
[28] S.D. Stoller, W. Happer, F.J. Dyson, Phys. Rev., A, 44, 7459 (1991) and references therein for a series of papers by G. Cates and W. Happer.
46
[29] G.P. Zientara, J.H. Freed, J. Chem. Phys., 72, 1285 (1980).
[30] H.C. Torrey, Phys. Rev., 104, 563 (1956).
[31] R. Kubo, M. Toda, N. Hashitsume, Statistica Physics, II, Springer-Verlag, New
York (1991).
[32] S. Axelrod, P.N. Sen (unpublished)
[33] W.T. Dixon, J. Chem. Phys., 77, 1800 (1982).
[34] O.N. Antzutkin, Progr. in NMR spectroscopy, 35, 203 (1999).
[35] G. Leu, X-W Tang, S. Peled, W. Maas, S. Singer, D.G. Cory, P.N. Sen, Chem.
Phys. Lett., 332:(3-4) 344 (2000).
[36] C.M. Reinstra, S. Vega, R.G. Griffin, J. Magn. Reson. A, 119, 256 (1996).
[37] J.A. Glasel, K.H. Lee, J. Am. Chem. Soc., 96, 970 (1974); P.N. Sen, S. Axelrod,
J. Appl. Phys., 86, 4548 (1999) and references therein.
[38] S.J. Lang, J. Magn. Reson. A, 104, 345 (1993).
47
48
Chapter 2
Two-dimensional exchange
diffusive magic angle sample
spinning
2.1
Detection of motion though susceptibility fields
in two-dimensional exchange diffusive-MASS
experiments
2.1.1
Summary
We show that the off-diagonal coherence peaks in two-dimensional Fourier Transform
NMR spectroscopy in fluids contained in porous media undergoing magic angle sample spinning (MASS) arise from amplitude modulation of the fluid’s magnetization.
The amplitude modulation originates from the combined effect of MASS and the
molecular diffusion through the inhomogeneous magnetic fields created by the susceptibility contrasts in the porous medium. The magnitude of the off-diagonal peaks
provides information on the porous medium’s structural length scales which give rise
to correlation length scales of the magnetic susceptibility.1
1
The material presented in this section is published in Chem. Phys. Lett. 336 588-594 (2002).
49
2.1.2
Background
The two-dimensional exchange experiment in NMR is a powerful probe of dynamics.
It consists of an encoding period, a mixing period and a detection period to record the
final state. In the encoding period the magnetization of each isochromat is labeled
by its resonant frequency. During the mixing period tm , the spins move, and the
frequency labels are redistributed during read out, and a two-dimensional Fourier
Transform reports on the correlation between starting and final states[1].
In studies based on MASS, the sample rotation introduces a frequency modulation
during the encoding. If the mixing time is not restricted to an integer multiple of the
period of rotation, an apparent new dynamics is introduced by this coherent spinning
[4]. Since this apparent dynamics has no additional information over the diagonal
spectra, MASS two-dimensional exchange measurements are invariably performed
with rotor synchronization [3, 5].
In this work, we explore the physics behind the appearance of off-diagonal peaks
in two-dimensional exchange spectra of fluids moving in the susceptibility field in the
interstitial space of complex matter in a rotor synchronized two-dimensional experiment.
The NMR spectra of fluids in porous and granular matter are generally broadened
by the range of local fields encountered due to spatial variation of the bulk magnetic
susceptibility across the sample. MASS is well known as a means of removing this inhomogeneous broadening and permitting high resolution NMR spectroscopy. MASS
is employed to average out the range of susceptibility fields and return the NMR
spectra to high resolution [4, 6]. In solids, where the spins do not move in the susceptibility field, MASS leads to the usual frequency modulation of the NMR response
[3, 4, 6].
For rapid diffusion, there is an amplitude modulation of the NMR response in
MASS [8, 9, 10]. Here we show that for rotor synchronized two-dimensional exchange
spectra, it is the amplitude modulation which leads to the appearance of off-diagonal
resonances, and we provide a quantitative description of their intensity. If molecular
50
(*),+
()+
%&
(*)+
tm
%#'
"!#" $
Figure 2-1: Pulse sequence of 2D-exchange experiments.
diffusion does not lead to a time-varying field, the rotor synchronized two-dimensional
exchange suppresses the off-diagonal resonances and a high resolution spectrum is
recorded along the diagonal. The amplitude modulation re-introduces the off-diagonal
resonances even in the case of rotor synchronization. The expression for the intensities
of these off-diagonal resonances provides information on the correlation length of
the susceptibility field, and, thus, ultimately provides information on the range of
feature length scales that are the source of the susceptibility spatial variation. Our
goal is to have an experimental handle on these length scales that characterize these
complex material, which are important in diverse fields ranging from oil recovery,
heterogeneous catalysis to biological perfusion.
2.1.3
Two-dimensional exchange in diffusive MASS
The 2D exchange pulse sequence is π/2 − t1 − π/2 − tm − π/2 − t2 and it is shown in
Figure 2-1. The first π/2 pulse prepares the magnetization in the transverse plane.
During the evolution time t1 , the spins are labeled by their local fields. The second
π/2 pulse stores the magnetization along the z-axis. After a mixing time tm , the spins
diffuse to new locations in the sample. The third pulse restores the magnetization to
the transverse plane. The spins then are relabeled by the new local fields. The signal
is recorded during the acquisition time t2 . The differences of the local fields between
the evolution time and acquisition time introduce the off-diagonal resonances. In
MASS 2D exchange experiment we use tm = n ω2πr , where ωr is the angular velocity
of the spinner. In this way, there are no extra dynamics introduced by the sample
spinning itself [5]. Note that diffusion is present at all times and gives a periodic
51
amplitude modulation term when the spin evolves during the evolution time and
the acquisition time [8, 10]. However, at high spinning frequencies, the phase term
dominates and we can neglect this part of diffusion effect.
The spin precession during t1 causes a time dependence on phase that reflects
the modulation by MASS. The magnetization after the encoding time, neglecting
diffusion, is
M (r, t1 ) = ρ(r)ei
R t1
0
γB(r,θ,ωr t+φ)dt
(2.1)
where ρ(r) is the initial spin density of the sample. The spins at different locations
gain different phases. Molecular diffusion during the subsequent mixing time tm
breaks the correlation between the spin’s phase and spatial location and results in
the amplitude modulation [8, 10].
The magnetization after the mixing time tm can be expressed as
M (r, t1 , tm ) =
R
drj ρ(rj )ei
R t1
0
γB(rj ,θj ,ωr t+φj )dt
P (rj , t1 |r, t1 + tm )
(2.2)
Here P (rj , t1 |r, t1 + tm ) is the conditional displacement propagator, which describes
the probability density of a spin at rj before the mixing time, diffusing to r after
the mixing time. The propagator description of molecular motion is complementary
to the time evolution operator method used in [8, 10]. Throughout, the standard
phase cycling procedure is used, so that we need to consider only the magnetization
in the x-y plane. The weak bulk and surface relaxation effects in the simple system
studied here are given by simple exponential terms for longitudinal and transverse
magnetizations, respectively. To reduce clutter, we do not display the relaxation
terms explicitly.
Ignoring the effect of diffusion during the acquisition time t2 , the magnetization
can be expressed as:
M (r, t1 , t2 , tm ) =
R
drj ρ(rj )e
i
R t1
0
γB(rj ,θj ,ωr t+φj )dt
P (rj , t1 |r, t1 + tm )e
i
R t1 +tm +t2
t1 +tm
γB(r,θ,ωr t+φ)dt
(2.3)
In porous media, if the ratio of the centrifuge force to the surface tension is small,
52
there is no coherent motion [9]. The sample spinning does not modify the propagator
from the static (non-spinning) propagator: P (rj , t1 |r, t1 + tm ) = P (rj , 0|r, tm ). With
the synchronization condition that the mixing time is an integer n multiple of the
rotor period, tm = n2π/ωr , the above equation becomes
M (r, t1 , t2 , tm ) =
R
drj ρ(rj )e
i
R t1
0
γB(rj ,θj ,ωr t+φj )dt
P (rj , 0|r, tm )e
i
R t1 +t2
t1
γB(r,θ,ωr t+φ)dt
.
(2.4)
To show the essential physics of the cross peaks, we separate the phase and the
amplitude parts,
M (r, t1 , t2 , tm ) = Aamp (r, t1 , tm )ei
Aamp (r, t1 , tm ) =
R
drj ρ(rj )ei
R t1
0
R t1 +t2
0
γB(r,θ,ωr t+φ)dt
γB(rj ,θj ,ωr t+φj )dt
P (rj , 0|r, tm )e−i
R t1
0
γ.B(r,θ,ωr t+φ)dt
(2.5)
The amplitude modulation (at the location r) Aamp (r, t1 , tm ) is due to the interplay
of the MASS and molecular diffusion. The observed signal is given by a spatial
integration of Eqs. (2.5).
Eqs. (2.5) show that it is the amplitude modulation that leads to the off diagonal
peaks. In absence of diffusion P (rj , 0|r, tm ) → δ(rj − r), and the amplitude term
becomes a constant. The phase term is independent of tm (tm being an integer
multiple of rotor period) and depends on t1 and t2 through the combination t1 + t2 .
Thus, in absence of diffusion, a two-dimensional Fourier transform of the signal (the
phase term by itself, which is a function of t1 + t2 ) with respect to t1 and t2 leads only
to diagonal peaks. These diagonal peaks are at integer multiples of rotor frequency.
The last conclusion follows from the fact that the local field is modulated periodically
at the rotor frequency and an expansion of the phase term in terms of the usual BesselFourier series [4] gives terms oscillating at integer multiples of the rotor frequency.
In the presence of diffusion, the amplitude term of Eqs. (2.5) gives rise to terms
that depend on t1 but not on t2 . This, together with the phase term, gives rise to
cross-peaks in a two-dimensional Fourier transform of the signal.
So far, except for the neglect of diffusion during encoding and acquisition periods,
our arguments have been exact. The expressions in Eqs. (2.5) above are rather com53
plex, and in order to make a comparison with experiments, judicious approximations
are required.
To get some useful results, consider short mixing times, such that, to the lowest
order in time, we may neglect the effect of geometrical restriction and the propagator
P (rj , 0|r, tm ) is assumed to be free diffusion. For such short periods, we may expand
the argument in Exp[B(rj , θj , ωr t + φj ) − B(r, θ, ωr t + φ)] in Eqs. (2.5) in a Taylor
series, and, performing a second cumulant expansion, the amplitude modulation can
be simplified,
Aamp (r, t1 , tm ) ≈ ρe
−Dtm (∇
R t1
0
γB(r,θ,ωr t+φ)dt)2
(2.6)
Here, ∇ is the gradient operator. This is completely equivalent to the second cumulant approximation of [8, 10]. We have assumed that the initial spin density ρ(r)
is independent of position, which is the case for most experiments such as those
described below.
Eq. (2.6) has a simple interpretation. The local frequency gradient γ∇B(r, θ, ω r t+
φ), after being integrated over time, corresponds to a wave-vector k = γ
R t1
0
∇B and
Eq. (2.6) represents a spatial Fourier transform (with respect to this wave-vector) of
the free diffusion propagator of duration tm , i. e., A ≈ Exp[−Dtm k2 ].
The exponent in Eq. (2.6) contains periodic functions of time ωr t1 . Thus, the
amplitude can be expanded, as noted above, in the usual Fourier-Bessel series as in [4]
with phases that are integer multiple of ωr t1 and with amplitudes that are modified
Bessel functions of the corresponding integer orders. The argument of the Bessel
function is of the order (∆ω/ωr )2 (LD /Lc )2 . Here ∆ω is the frequency broadening due
√
to susceptibility and Lc is the characteristic length over which it varies. LD = 2D0 tm
is the diffusion length in the mixing time and provides an additional length scale
which is under the control of the experimenter through the choice of the mixing
times. Information of internal gradients and correlation lengths may be obtained
through the analysis of the change in side-bands with the change of the mixing time
tm .
54
Diffusion during t1 (encoding) and t2 (acquisition) were neglected to make the
physics of the origin cross-peaks clear. During these two periods, the spreads in the
acquired phases make the phase terms far more important than the corresponding
amplitude terms. Inclusion of diffusion will add, in addition to decay, amplitude modulations that are periodic functions of t1 and t1 + t2 , and, thus, will contribute to the
cross-peaks. Furthermore, the effects of geometrical restriction will be non-negligible
when these times are large such that the diffusion length becomes comparable to
about a quarter of the bead-size.
To make comparison with experiments on bead packs filled with water, we next
compute the local field. Local magnetic fields of porous and granular systems, with a
weak susceptibility contrast, can be approximated by a superposition of dipole fields
[7] of individual grains. However, the essential physics at short times can be obtained
by considering the dynamics of a spin in the field of a single dipole. This is due to
the fact that the gradients are the strongest near the grain-fluid interface and in a
short time the spins do not have a chance to leave this region.
The local filed (along z-axis) of a dipole with magnetic dipole moment, M, polarized along z axis, is:
B(r) =
M
(1 − 3cos2 β)
|r|3
(2.7)
Here r is the vector from the dipole center to the spin, and β is the angle between
B0 = B0 ẑ and r.
This field can be written in the rotor frame, which spins with the rotor at a
frequency ωr and is tilted at the magic angle,
B(r, t) =
´
√
M ³
2
cos(2
φ
+
2
ω
t)
sin
θ
+
2
cos(φ
+
ω
t)
sin(2
θ)
r
r
r3
(2.8)
or as
B(r, t) = c1 cos(ωr t) + c2 cos(2ωr t) + s1 sin(ωr t) + s2 sin(2ωr t),
(2.9)
where, θ is the angle between the rotating axis and the vector from the dipole center
55
to the spin, and φ is the orientation of the rotor along the spinning axis, and c 1 =
√
√
(M/r3 ) 2 sin(2θ) cos(φ), c2 = (M/r 3 ) sin2 θ cos(2φ), s1 = −(M/r 3 ) 2 sin(2θ) sin φ,
s2 = −(M/r 3 ) sin2 θ sin(2φ), are functions of position.
The amplitude modulation can be obtained by directly substituting the dipole
magnetic field formula from Eq. (2.9) into the above Eq. (2.5) to give
2
Aamp (r, t1 , tm ) = ρe
− Dtm2γ (∇c1 sin(ωr t1 )+
ωr
∇c2
2
sin(2ωr t1 )+∇s1 (1−cos(ωr t1 ))+
∇s2
(1−cos(2ωr t1 )))2
2
(2.10)
While the phase in Eq. (2.5) is a periodic function of ωr (t1 + T2 ), the amplitude
is a periodic function of ωr (t1 ) and a two-dimensional Fourier transform with respect
to t1 , t2 will contain off-diagonal peaks. To estimate quantitatively the magnitude of
the peaks, we replace the square of field gradients by the average of the squares– a
mean field approximation that often works well. Formally this result is derived by
performing a cumulant expansion to the lowest order in gradient [8, 11] and taking
a spatial average around a sphere. As the susceptibility field, Eq. (2.7), falls off as
the cube of the distance from the center of a sphere, it suffices to integrate from the
surface of the sphere (radius a) to twice the radius where the field gradient is down
by a factor of 16. Combining Eq. (2.9) and Eq. (2.10), and after an angular average,
a second second-cumulant expansion gives
< Aamp (t1 , tm ) >≈ ρe
93Dtm γ 2 M2
(−9+8 cos(ωr t1 )+cos(2ωr t1 ))
2
560a8 ωr
.
(2.11)
Equation (2.11) contains an oscillatory part, whose amplitude is proportional to
(∆ω/ωr )2 (Dtm /L2c ). Here Lc is a and ∆ω ≈ γM/a3 .
2.1.4
Experimental results and discussions
We next describe rotor synchronized two-dimensional experiments. The sample is
made of 50µm glass beads packed in a 4mm MASS Zirconia rotor filled with water
or wax. All the spectra were taken at 200 C and at 500 MHz using a Bruker DRX500
56
9
(kHz)
7
5
3
1
1
3
5
(kHz)
7
9
Figure 2-2: MASS 2D exchange spectrum of wax embedded in a pack of beads
(tm =30ms)
spectrometer equipped with a high resolution 1 H/13 C MASS probe with gradient coils.
The results presented here are for the rotor frequency of 2kHz.
A static proton spectrum of water in the pack of glass beads shows a single broad
resonance, of width of about 2kHz. Since the mixing time is 5ms, the diffusion length
is much smaller than the bead size. In order to have the static linewidth in a solid
similar to that of water in 50 µm glass beads, we used a wax – poly( ethylene glycol)
to replace water in 50 µm glass beads [10]. The wax spectrum (see Fig. 2-2) has
only peaks along the diagonal. The water spectrum (see Fig. 2-3) shows clearly the
existence of off-diagonal resonances, although the mixing time tm = 5ms is much
shorter than the mixing time of the experiment on the wax sample tm = 30ms.
We made the mixing time in the water experiment relatively short tm = 5ms to
minimize the effects of restriction on diffusion during this period. For the wax sample,
experiments with tm = 5ms show no evidence of diffusion, and not even when we
make it as long as tm = 30ms. The intensity of the cross peaks is determined by the
interplay of the frequency modulation and the amplitude modulation.
To emphasize the cross-peaks, we project the two-dimensional spectrum on to a
line which is rotated 450 with respect to the ω2 -axis. This is obtained by integrating
57
9
kHz
7
5
3
1
1
3
5
7
kHz
9
Figure 2-3: MASS 2D exchange spectrum for water embedded in a pack of beads
(tm =5ms)
1
1
0.8
0.6
0.4
0
5
0
kHz
0.2
-5
0
4
2
0
kHz
-2
-4
Figure 2-4: The 45o integration spectrum of 2D spectrum of wax. The insert is the
slice spectrum along the diagonal line of the 2D spectrum of wax.
58
1
1
0.8
0.6
0.4
0
5
0
kHz
0.2
-5
0
4
2
0
kHz
-2
-4
Figure 2-5: The 45o integration spectrum of 2D spectrum of water in bead pack. The
insert is the slice spectrum along the diagonal line of the 2D spectrum of water
the two-dimensionsl spectrum along the lines that are parallel to the main diagonal
ω1 = ω2 (Figs. 2-4 and 2-5). If there is no diffusion, D = 0, the amplitude modulation remains constant and all the resonances are along the diagonal line of the
two-dimensional spectrum. Therefore, the integrated spectrum will show a side-band
free one-dimensional spectrum, as seen for the wax spectrum in Fig. 2-4. The diffusion introduces the cross peaks, which result in the side-bands in the one-dimensional
integration spectrum as demonstrated in Fig. 2-5. The intensities of the first and
second side-bands of the integrated water spectrum are listed in Table 2.1.
A crude estimate of the ratio of diffusional side-bands can be obtained from Eq.
(2.11) as follows. Since the exponent is rather small, expanding and keeping terms
linear in it gives an amplitude that is proportional to 8 cos(ωr t1 ) + cos(2ωr t1 ) which
predicts a ration of 8:1 for the first to the second side-band. The experimentally
observed ratio of the first to second side-bands intensities is about 7:1 (with an estimated error of about ±20%), see Table 2.1. This is not too surprising. Since dipole
fields decrease as 1/r 3 , a spin mainly feels the field of single sphere.
59
Table 2.1: Sideband intensities of the integrated spectrum along diagonal line of the
2D spectrum in the sample with water in a pack of beads.
Sideband Order
Intensity
2.1.5
2
0.0045
1
0.031
0
-1
1 0.035
-2
0.0048
Conclusions
In conclusion, the two-dimensional exchange experiment of fluids in porous media
carries information about the dynamics of the molecular motion and the landscape,
i.e. the restricted geometry and field inhomogeneity, over which this motion is taking
place. Off diagonal peaks are present in the two-dimensional exchange spectra in the
presence of diffusion. Two-dimensional exchange makes the amplitude term of the
complex magnetization depend on the azimuthal angle φ + ωr t1 and the phase term
as φ + ωr (t1 + t2 ). These give terms periodic in ωr t1 and ωr (t1 + t2 ) and thus both
diagonal and off-diagonal peaks appear in the two-dimensional Fourier transform of
the signal with respect to t1 and t2 .
The molecular motion reports on the correlation lengths of the susceptibility gradients which are closely related to structure. Two-dimensional exchange provides
a convenient and powerful method for focusing at the amplitude modulation and
the information on the correlation lengths contained there in. This work introduces
the physics of probing the dynamics from off-diagonal resonances of two-dimensional
exchange spectra and sets up method of analyzing the data.
60
2.2
Selectively observing the amplitude modulation under MASS
Magic angle sample spinning (MASS) averages the inhomogeneous magnetic field due
to the susceptibility contrast in porous media by modulating the local magnetic field
(frequency modulation). Molecular diffusion introduces a homogeneous broadening,
which modulates the amplitude of the signal (amplitude modulation). The depth of
amplitude modulation is determined by the interplay of molecular diffusion and MASS
averaging and contains rich information on local magnetic fields, through which the
spatial structure of the sample may be obtained. In this section, we present two methods to quantify the amplitude modulation: a phase suppressed method and a more
conventional two-dimensional exchange method. The phase suppressed method directly observes the amplitude modulation in the time domain. The approximate equations are derived to extract the physical information from the time domain data. The
conventional two-dimensional exchange spectrum contains both the frequency modulation and the amplitude modulation terms. The amplitude modulation introduces
cross peaks in the two-dimensional spectrum. The integration of the two-dimensional
spectrum along lines parallel to the main diagonal line will give a one-dimensional
spectrum, that is the Fourier Transform of the amplitude modulation.2
The goal of this work is to investigate methods of nuclear magnetic resonance
(NMR) appropriate to the study of fluid dynamics inside porous media by the magic
angle sample spinning (MASS).
As a non-invasive method, NMR is a powerful tool to study fluid dynamics in
porous media [12]. However, the susceptibility difference inside the sample introduces
strong local magnetic fields, which broadens the NMR spectra [7]. MASS averages out
this broadening effect [6]. This is a standard technique in modern solid state NMR
employed to average out the broadening effect from the chemical shift anisotropy to
obtain high resolution spectra [4]. In porous media, the local field introduced by the
susceptibility contrast can be approximated by a superposition of dipolar fields and
2
The material presented in this section will be published in J. Chem. Phys. (2003).
61
can be averaged out by MASS. The MASS spectrum consists of a center peak flanked
with sidebands, each of which has a much narrower line width compared to the static
spectrum. The faster the spinning, the smaller the amplitude of the sidebands.
However, molecular diffusion introduces the complication that the spins move
through the susceptibility fields and thus, as described in previous papers,[8, 10, 13]
the NMR signal contains two components: an amplitude modulation due to the
homogeneous broadening effect, and a frequency modulation part due to the inhomogeneous effect. The frequency modulation only contains the static field information,
while the amplitude modulation, which is caused by the interplay of the molecular
diffusion and the MASS, can be used to probe fluid dynamics in porous media. From
the later the structural information can be obtained, such as characteristic length
scales [8, 10, 13].
In order to quantify this dynamics, it is useful to separate the amplitude modulation from the frequency modulation. Here we discuss two methods, which can be
used to separate the amplitude modulation and the frequency modulation and show
how to extract useful physical information.
For the two-dimensional exchange method (Figure 2-1), the magnetization of the
sample during the acquisition time can then be written as [13]:
M (t1 , t2 , ωr ) =
R
Aamp (r, t1 , ωr ) =
drAamp (r, t1 , ωr )Af (r, t1 + t2 , ωr )
R
drj ρ(rj )e
Af (r, t1 + t2 , ωr ) = eiγ
R t1 +t2
0
iγ
R t1
0
dtB(rj ,t,ωr )
P (rj , 0|r, tm )e
−iγ
R t1
0
dtB(r,t,ωr )
(2.12)
dtB(r,t,ωr )
where γ is the gyro-magnetic ratio, ωr is the spinning frequency, B(rj , t, ωr ) is the
magnetic field at position rj at time t with the spinning frequency ωr , P (rj , 0|r, tm )
is the conditional displacement propagator, which describes the probability density
of a spin at rj before the mixing time, diffusing to r after the mixing time. Here,
Aamp (r, t1 , ωr ) is the amplitude modulation part due to the interplay of the molecular
diffusion and the MASS, while Af (r, t1 +t2 , ωr ) is the frequency modulation part due to
the MASS. Since the local gradient is position dependent, the amplitude modulation
varies with location.
62
The frequency modulation contains information only on the static internal field.
The amplitude modulation contains dynamic information of fluids inside samples and
can be used as a probe to observe the dynamics. As they are generally entangled with
each other, extra steps are needed to be taken to separate them. Here we discuss two
methods: phase suppressed method and two-dimensional exchange sequence method.
2.2.1
Phase suppressed method
One way to observe the amplitude modulation term is to directly suppress the frequency modulation term.
In order to suppress the frequency modulation term, let’s first analyze the properties of the internal magnetic fields. Since in the rotor frame B(r, t, ω r ) is a periodic function of time, when t1 + t2 is an integer multiple of the rotation period τr ,
Af (r, t1 + t2 , ωr ) = 1. This can easily be implemented by modifying the basic stimulated echo sequence to
π
2
- te -
π
2
- tm -
π
2
0
- tw - t2 , where, te = τr + t1 , is the encoding
0
time; tw = nτr −t1 , is the waiting time before the acquisition, and t2 is the acquisition
time.
0
For every t1 , when t2 = 0, te + tw = nτr so that
magnetization is
M (t1 , ωr ) =
=
R
R
R nτr
0
dtB(r, t, ωr ) = 0. The
drAamp (r, t1 , ωr )
dr drj ρ(rj )eiγ
R
R t1
0
dtB(rj ,t,ωr )
P (rj , 0|r, tm )e−iγ
R t1
0
dtB(r,t,ωr )
(2.13)
.
0
Therefore, the time domain data at t2 = 0 gives the ensemble amplitude modulation
of the sample.
In porous media, made of discrete spherical particles, the local magnetic field at
the position r can be approximated by the superposition of dipole fields generated by
discrete sources:
B(r) =
P
Mk
k |r−rk |3 (1
− 3cos2 βk (r))
(2.14)
where rk is the position of the kth dipole, i.e. the center of the kth sphere, Mk is
the magnetic dipole moment of the kth dipole and βk (r) is the angle between B0 and
63
r − rk . This field can be written in the rotor frame, which spins with the rotor at a
frequency ωr and is tilted at the magic angle, as
B(r, t, ωr ) =
Mk
(cos(2 φk (r) + 2 ωr t) sin2 θk (r)
3
|r
−
r
|
k
k
√
+ 2 cos(φk (r) + ωr t) sin(2 θk (r)))
X
(2.15)
where θk (r) is the angle between the rotating axis and the vector r − rk , and φk (r) is
the orientation of the vector r − rk along the spinning axis in the rotor frame.
The amplitude modulation bears all the fingerprints of the space structures that
the spins can tell us as they diffuse through the sample. In order to extract useful
physical information from the expression (2.13), some additional assumptions need
to be made about the propagator. In randomly packed samples, we can assume that
the propagator depends on the difference of the azimuthal angle, ∆φ, instead of the
absolute value of the azimuthal angle, φ.
In general, we can expand e
iγ
R t1
0
dtB(rj ,t,ωr )
and e
−iγ
R t1
0
dtB(r,t,ωr )
in Bessel functions
and analyze the intensity of the different terms. In order to obtain quantitative physical information, we spin the sample to a moderately high spinning frequency so that
only one or two side-bands appear in the spectrum. By doing this, we can use perturbation theory to analyze the signal. Spinning faster would make the approximation
better, however, if the spinning frequency is too fast, the internal field will be totally
averaged out by MASS and diffusion would give us no information on internal fields.
The phase term before the mixing time from Eq. (2.15) is:
Φ(rj , t1 , ωr ) = γ
=
=
R t1
0
dtB(rj , t, ωr )
R t1 P
0
k [c1k (rj )cos(φk (rj )
+ ωr t) + c2k (rj )cos(2φk (rj ) + 2ωr t)]
c1k (rj )
k [ ωr (sin(φk (rj ) + ωr t1 ) − sinφk (rj ))
+ c2k2ω(rrj ) (sin(2φk (rj ) + 2ωr t1 ) − sin(2φk (rj )))]
P
where
c1k (rj ) =
c2k (rj ) =
Mk
|rj −Rk |3
√
2sin(2θk (rj ))
Mk
sin2 (2θk (rj ))
|rj −Rk |3
64
.
(2.16)
(2.17)
Similarly,
Φ(r, t1 , ωr ) = −γ
= −
R t1
P
0
n[
dtB(r, t, ωr )
c1n (r)
(sin(φn (r)
ωr
+ ωr t1 ) − sinφn (r))
(2.18)
(r)
(sin(2φn (r) + 2ωr t1 ) − sin(2φn (r)))] .
+ c2n
2ωr
By expanding ei(Φ(rj ,t1 ,ωr )−Φ(r,t1 ,ωr )) in a Taylor series, keeping terms only up to
the second order, we can obtain the approximative result
M (t1 , ωr ) ≈ g3 (1 − (g1 + g2 ) + g1 cos(ωr t1 ) + g2 cos(2ωr t1 ))
(2.19)
where
g3
g1
g2
∆φk,n
=
Z
dr
Z
drj ρ(rj )P (rj , 0|r, tm )
+ n c21n (r) − 2 k,n c1k (rj )c1n (r)cos(∆φk,n ))
=
ωr2 g3
R
R
P 2
P
P
dr drj ρ(rj )P (rj , 0|r, tm )( k c2k (rj ) + n c22n (r) − 2 k,n c2k (rj )c2n (r)cos(2∆φk,n ))
=
4ωr2 g3
= φk (rj ) − φn (r) .
R
dr drj ρ(rj )P (rj , 0|r, tm )(
R
P
2
k c1k (rj )
P
P
Although we obtain Eq. (2.19) by Taylor series expansion, the same equation can
be obtained by Bessel function expansion. However, g1 , g2 and g3 will be functions
of complicated Bessel functions which obscures the physical meaning. We point out
that the validity of Eq. (2.19) depends on how well we can ignore the higher order
terms and the Taylor series will put stricter requirement on the exponent compared
with the Bessel function expansion. For example, if the mixing time is very long, so
that the diffusion length of the spins is comparable to the characteristic length scale
of the sample, then the Taylor series expansion is good provided that the spinning
frequency is about two times larger than the static spectral line-width of the sample,
while the Bessel function expansion method only requires that the spinning frequency
is larger than the line-width of the static spectrum of the sample.
65
2.2.2
Two-dimensional exchange method
Second approach to separate the frequency and amplitude modulation is directly to
use the two-dimensional exchange sequence. This sequence is widely used to investigate chemical shift exchange phenomena. It is also a powerful experiment for studying
diffusion phenomena. Compared to the phase suppressed experiment, it contains more
information since it records both the frequency and amplitude modulation. We have
presented this briefly in Ref. [13].
Equation (2.12) is the general expression of the signal for the two-dimensional
exchange experiment. Following the two-dimensional Fourier Transform, there are
resonances on both the diagonal line and off-diagonal places if diffusion is present
[13]. Notice that if there is no diffusion, the propagator is a δ function, and then
Aamp = 1 and M =
R
drk Af (t1 + t2 ). The dependence of M on t1 + t2 results in the
two-dimensional Fourier Transform only having resonances on the diagonal line, i.e.,
the two-dimensional Fourier Transform of the frequency modulation Af only appears
on the diagonal. However, if there is diffusion, the amplitude modulation A amp will
depend on t1 . The Fourier Transform of the amplitude modulation convolved with
diagonal peaks generated by the frequency modulation introduces the off diagonal
peaks.
Therefore, the two-dimensional spectrum contains both the amplitude and frequency modulation information. In order to effectively “deconvolve” the spectrum
to extract the amplitude modulation, it is sufficient to integrate the 2D spectrum
along lines parallel to the main diagonal. In this section, we prove this statement
mathematically and show a second moment analysis method to extract physical information.
Denote the one-dimensional Fourier Transform of Af (t) as Af (ω) and the onedimensional Fourier Transform of Aamp (t) as Aamp (ω). From Eq. (2.12), the twodimensional Fourier Transform of the whole signal can be expressed as:
M̃ (ω1 , ω2 ) =
R
drÃamp (ω1 , ω2 ) ⊗ Ãf (ω1 , ω2 )
= Aamp (ω1 )δ(ω2 ) ⊗ Af (ω2 )δ(ω1 − ω2 )
66
(2.20)
where ⊗ means convolution.
The integration along the lines parallel to the diagonal gives
R∞
−∞
√
√ R
d( 2ω2 )M̃ (ω1 , ω2 ) |ω1 −ω2 =ω0 = 2 drAamp (ω0 ) .
Here we used the fact that
R
(2.21)
dωAf (ω) = Af (t) |t=0 = 1.
For the preliminary analysis, a moment analysis was used to extract physical
information from the 1D spectrum obtained by integration. We show that the second
moment of the Fourier spectrum of the amplitude modulation shows the correlation
of the magnetic field in the system correlated by the propagator.
Denote the second moment as M2 .
M2 =
=
R
R
2
dωω 2 Aamp (ω) = − dtd 2 Aamp (t) |t=0
drj (γB(rj , ωr , t))2 + dr(γB(r, ωr , t))2
−2
RR
R
(2.22)
drj dr(γB(rj , ωr , t))(γB(r, ωr , t))P (rj , 0|r, tm ) |t=0
Both methods, the phase suppressed method and two-dimensional exchange method,
therefore, can detect the amplitude modulation without the complication from the
frequency modulation. The phase suppressed method is more time efficient. However
the conventional two-dimensional exchange method contains richer information since
it simultaneously obtains both the frequency modulation and amplitude modulation.
2.2.3
Experimental results of the phase suppressed method
All our experiments are performed on a Bruker DRX500 spectrometer equipped with a
high resolution 1 H/13 C MASS probe with gradient coils. The samples used are packed
glass beads filled with 1% water and 99% D2 O. The large amount of deuterated water
minimizes the radiation damping effect. By changing the size of glass beads, we can
control the local gradients and characteristic length scale.
Figure 2-6 shows the experimental results from a sample of 50µm diameter glass
beads. The different curves in the figure correspond to different mixing times. t1 was
varied from 0 to one rotating period. The experimental data is normalized to remove
67
&(
%! '
#
&"
#%$
! " Figure 2-6: Results of phase suppressed experiments of water in packed 50µm glass
beads showing the dependence of the ensemble amplitude modulation on t1 (Eq. 2.13)
for three values of the mixing time (tm = 10ms, 100ms and 1000ms), at a spinning
frequency of 2kHz. The data for each experiment was normalized with respect to the
first data point. The curves obtained by fitting the experimental data points with
Eq. 2.19 are shown as solid lines.
68
Table 2.2: Fitting results for different mixing times. The sample is the packed 50µm
glass beads filled with 1% water and 99% D2 O.
Mixing Time (ms)
g1
g2
10
0.1417
0.0342
100
0.2570
0.0847
500
0.2991
0.1060
1000
0.3111
0.1126
the effect of T1 relaxation. The fitting results are listed in Table 2.2.
g3 is not particularly interesting since g3 should be around one due to the data
normalization (neglecting relaxation) and it is not listed out in the table. The values of
g1 and g2 increase with the increase of the diffusion length by increasing the mixing
time. This is expected since the longer the diffusion length, the farther away a
molecule diffuses and the final field that the spin feels is less correlated with the
original field. By observing g1 and g2 , the information on the gradient of the local
field can be obtained.
In Eq. (2.19), by taking ωr t1 = 2nπ with n = 0, 1, ..., the total magnetization
M equals g3 . If the encoding time is chosen such as ωr t1 = (2n + 1)π, the total
magnetization M equals g3 (1 − 2g1 ). By combining these two, we can directly obtain
g1 rather than by fitting the curve of many experimental points over a rotating period,
although the fitting method is more accurate.
Figure 2-7 shows the change of g1 when the diffusion distance increases for differ√
ent samples. The diffusion distance is defined as 2Dtm , where D is the free diffusion
coefficient and tm is the mixing time. The spinning frequency used in these experiments was 5kHz. Sample A corresponding to the top curve is packed with 50µm
spherical glass beads. Sample C corresponding to the bottom curve is packed 100µm
beads. Sample B, corresponding to the middle curve, is made of glass beads with
two different sizes, 50µm and 100µm, which are packed in separate regions inside the
rotor (the 50µm glass beads occupy the bottom half of the rotor and the 100µm beads
occupy the top half) and are not mixed together. The static line width of sample A
is about 2.5kHz. The static line width of sample C is about 2.1kHz. The static line
width of sample B is between 2.1kHz and 2.5kHz. Due to the larger bead size, the
gradient inside the sample C is much smaller than the gradient field inside the sample
69
,-
.0/
476/ 898:;<6=>8@?BADC
.E/ µ1325/>
µ
µ1325476898:;<6=58I? JKC
G
1
9
F
H
/>/
H
µ1325476898:
;6<=58I?LMC
#
! "$!%
& ')(
µ*)+
Figure 2-7: Increase in g1 with the diffusion distance in samples with different bead
sizes. All the samples are packed glass beads filled with 1% water and 99%D2 O. The
sample corresponding to the middle curve is packed one half with 50µm glass beads
and one half with 100µm glass beads. The spinning frequency in all the experiments
is 5kHz.
A.
2.2.4
Experimental results of the two-dimensional exchange
method
The Fourier Transform spectrum for a two-dimensional exchange experiment shows
both diagonal resonances and off-diagonal resonances. To separate the amplitude
modulation from the frequency modulation, we integrate spectrum along lines parallel
to the main diagonal.
Figure 2-8 shows a train of amplitude modulation spectra for different mixing
times from a sample of 50µm glass beads filled with 1% water and 99% D2 O. The
spinning frequency for all the experiments in Figure 2-8 is 2kHz. The center peak of
each spectrum is normalized as one. The normalization removes the T1 and intrinsic
T2 relaxation effects. Thus the change of the sidebands intensity is purely due to the
diffusion effect. The text beside each line is the mixing time used for that experiment.
70
Figure 2-8: The Fourier Transform of the amplitude modulation for different mixing
times obtained by two-dimensional exchange method in 50µm glass beads sample
filled with 1% water and 99% D2 O. The center peak of each spectrum is normalized
as one. The spinning frequency is 2kHz.
The symmetry of the sidebands in the amplitude modulation reflects the symmetry of the propagator, i.e., the probability for a spin diffusing from r 1 to r2 is the
same as the probability for a spin diffusing from r2 to r1 . The sidebands intensity
increases when the mixing time increases, which reflects the change of the propagator
for different mixing times.
As shown in the theory part, the second moment, M2 , of the spectrum of the
amplitude modulation reflects the correlation of the magnetic fields in the system
correlated by the propagator. In Eq. (2.22), as the mixing time becomes longer and
longer, the field will become more and more uncorrelated and the cross terms will
become smaller and smaller so that the second moment will become larger and larger.
Because the second moment M2 is proportional to the susceptibility difference
square, ∆χ2 , by removing the effect from the susceptibility difference, the change
of M2 will reflect the field change ratio, which is directly related to the curvature
information.
71
!"$#&%''()*(,+.-%0/1%02
354 6 4,4 µ7 4,4 µ µ
Figure 2-9: The normalized second moment RM2 (the ration of the second moment of
the amplitude modulation spectrum to the second moment of the Fourier Transform
of the FID) for different glass beads collapses when plotted as a function of scaled
diffusion length.
In order to obtain the susceptibility difference, we integrate the 2D-exchange spectrum along ω1 dimension. The resulted spectrum is actually the Fourier Transform
of a FID data after a single 90o pulse. The second moment of this spectrum, M2f id ,
is also proportional to ∆χ2 . Hence, the ratio, RM2 =
M2
,
M2f id
is independent of the sus-
ceptibility difference. For samples with different size of glass beads, the pore shape
√
m
geometry is similar except for a scale constant. RM2 will be a function of 2Dt
,
a
√
where 2Dtm is the diffusion length and a is the diameter of the glass beads. If the
susceptibility for one kind of glass beads is uniform, it can be shown that RM2 (
√
2Dtm
)
a
is independent of the glass beads size. This is confirmed by the experimental results
in Figure 2-9.
However, since the glass beads are not perfectly round, this introduces some irregular pore shapes, which accounts for the small discrepancies.
72
ω Figure 2-10: The change of g1 with different spinning frequencies.
2.2.5
Validity of the approximation for the phase suppressed
method
In the theoretical analysis for the phase suppressed method, we used perturbation
theory. Here we investigate the conditions of validity of the perturbation theory.
The sample we used is water in 50µm glass beads. The line width of the static
spectrum is about 2.5kHz. The mixing time is 1s, which is in the long mixing time
extreme for a pack of 50µm glass beads filled with water.
From theory, g1 and g2 are proportional with
show the experimental relation of g1 and g2 to
1
.
ωr 2
Figure 2-10 and Figure 2-11
1
.
ωr 2
The linear relation is quite good for g2 when the spinning frequency is greater
than 2kHz. However, g1 varies linearly with
1
ωr 2
only above 4kHz.
To evaluate the experimental change of g1 versus
1
ωr2
more quantitatively at lower
spinning frequencies, instead of Taylor series expansion, we take the Bessel expansion
and assume that a spin mainly feels the dipole field from a single glass bead and
that the spin distribution is spherical symmetric. The measured spectral line width
is about 2.1kHz. By applying these assumptions, we can calculate the theoretical
73
ω Figure 2-11: The change of g2 with different spinning frequencies.
dependence of g1 on
1
.
ωr2
This calculated theoretical curve is the solid line along the
experimental data points in Figure 2-10. We see that although our assumption is
rather coarse, the fits from Bessel expansion are reasonably good over the experimental frequency range.
To summarize, in the extreme of long mixing times, the approximate Eq. (2.19)
is valid when the spinning frequency is larger than the static line width. The Taylor
series expansion approximation of g2 is also good. The Taylor series expansion approximation of g1 needs to be justified by the Bessel function expansion. However, if
the spinning frequency is larger than two times of the static line width, the Taylor
series expansion approximation is good for both g1 and g2 analysis.
2.2.6
Conclusions
In this chapter, we explored two different methods to separate the amplitude modulation and frequency modulation due to diffusion under MASS. The amplitude modulation is determined by the spinning frequency, the propagator and local fields and
thus carries a lot of important information about the sample, such as the restricted
74
geometry and field inhomogeneity over which the molecules diffuse.
The phase suppressed method can directly observe the average amplitude modulation across the samples. In our experiments, when the samples are spun two times
faster than the static line width, the analytical approximation becomes simple, as
shown in Eq. (2.19) and the physical meaning for the different coefficients can be easily understood. It is shown that at short mixing times, g1 and g2 reflect the gradient
information in the sample.
The two-dimensional exchange experiment which gives a two-dimensional spectrum contains both frequency modulation and amplitude modulation information.
By integrating along the lines parallel to the main diagonal line, the amplitude modulation is separated from the frequency modulation. The second moment of the
Fourier Transform of the amplitude modulation is also related to the length scale
information.
75
76
Bibliography
[1] J. Jeener, B.H. Meier, P. Bachmann, R.R. Ernst, J. Chem. Phys., 71(11), 4546
(1979).
[2] L. Elmsley, A. Pines, Lectures on Pulsed NMR, second edition, Proc. Fermi
School, Soc. Italiana di Fisica, CXXIII (1994).
[3] K. Schmidt-Rohr, H.W. Spiess, Multidimensional Solid-state NMR and Polymers, Academic Press, New York (1994).
[4] M. Matti Maricq, J.S. Waugh, J. Chem. Phys., 70(7), 3300 (1979).
[5] A.P. Kentgens, E. de Boer, W.S. Veeman, J. Chem. Phys., 87, 6859 (1987).
[6] T.M. de Swiet, M. Tomaselli, M.D. Hürlimann, A. Pines, J. Magn. Reson., 133,
385 (1998).
[7] P.N. Sen, S. Axelrod, J. Appl. Phys., 86(8), 4548 (1999).
[8] G. Leu, X.-W. Tang, S. Peled, W.E. Maas, S. Singer, D.G. Cory, P.N. Sen, Chem.
Phys. Lett., 332, 344 (2000).
[9] G. Leu, A. G. Guzman-Garcia, D.G. Cory, P.N. Sen, Petrophysics, 43(1), 13
(2002).
[10] Y. Liu, G. Leu, S. Singer, D.G. Cory, P.N. Sen, J. Chem. Phys., 114, 5729
(2001).
[11] S. Axelrod, P.N. Sen, J. Chem. Phys., 114, 6878 (2001).
77
[12] C. S. Johnson, Jr, Diffusion Measurements by Magnetic Field Gradient Methods,
Encyclopedia of nuclear magnetic resonance, (John Wiley, New York, 1996), p.
1626.
[13] P.N. Sen, Y. Liu, G. Leu, D.G. Cory, Chem. Phys. Lett., 366, 588, (2002).
78
Chapter 3
Two-dimensional NMR-DQF
studies of heterogeneous samples:
quantitative characterization of
elastomer-carbon black interactions
3.1
Summary
A new method based on two-dimensional NMR double quantum filtering (DQF)
is used for the quantitative characterization of interactions between components in
biphasic heterogeneous media. The method is developed with the goal of being directly applicable to studies of rubber-like materials composed of two principal components: a relatively mobile phase made of elastomers and a more rigid phase made
of filer particles, usually carbon black (CB). The presence of filler particles induces
local susceptibility fields while the local ordering of the elastomer introduces dipolar fields. The separation of susceptibility and dipolar interactions allows for the
direct observation of three elastomer components near the surface of filler particles:
(1) encapsulated, (2) ordered on the filler surface, and (3) entangled with the surface component. The elastomer-filler interactions are responsible of enhancing the
79
physical properties of heterogeneous, rubber-like materials. Our method creates the
possibility to quantify the amounts of elastomer in each of these regions and to estimate the thickness of the surface component. The method is applied to the study of
several rubber samples for which the actual CB particle size distributions are determined from Scanning Electron Microscopy (SEM) measurements. For these samples
the length scale of the surface elastomer component is estimated and it is shown that
the encapsulated elastomer is most probably located between the graphitic plates of
the CB particles rather than in the space between adjacent particles.
3.2
Background
The special physical and chemical properties of rubber make it one of the most important materials employed in various industries. The rubber samples form a multicomponent, complex system made of carbon black particles (filler) and polymer chains.
The long polymer molecules are responsible for its specific extensibility and elasticity
and the filler particles (in most cases carbon black) complement these properties.
This process, known as reinforcement (i.e. the compounding of fillers with elastomers) is commonly employed for manufacturing materials with improved physical,
mechanical or electrical properties [1, 2, 3, 4]. The complex carbon black-polymer and
polymer-polymer interactions are responsible for the useful range of physical properties of rubber such as abrasion resistance, tensile stress at brake, tear propagation
resistance, etc.
Despite numerous investigations on carbon black filled rubbers using different techniques such as dynamic, mechanical, thermal and sorption analysis and spectroscopic
methods, the molecular origin of the reinforcement effect is still under discussion
[1, 2, 5, 6]. The NMR spectroscopic techniques can also be used to study the matrixfiller interphase (e.g. assessment of the degree of immobilization of rubber chains,
extent of layer thickness, etc.) [7, 8, 9, 10, 11, 12, 13, 14].
The main goal of our study is to develop a new method using two-dimensional
NMR experiments for investigating the complex interactions in rubber-like materials
80
composed of a mobile polymeric phase and a more rigid phase made of filler particles (usually CB). We are using the local susceptibility fields that appear at the
filler-polymer interface to investigate the structure and composition of these types of
samples. We apply this method to the study and quantitative characterization of the
CB-elastomer interactions in several rubber samples provided by Goodyear Research.
It has been theorized that there are three elastomer components in the rubber
samples distinguished by their characteristic molecular mobility [9, 15, 16]: immobilized fraction, intermediate fraction and mobile fraction. The immobilized fraction is
the elastomer that is immobilized on the carbon black surface, trapped in the carbon black agglomerates such that all the motions are restricted. The intermediate
fraction is the elastomer very close to the surface of the carbon black particles, such
that its dynamics is influenced by the presence of the carbon black particles. The
mobile fraction is further from the carbon black particles such that the motions are
not restricted and has the characteristics of bulk elastomers. Our method allows for
the observation and measurement of the properties of these components directly by
NMR.
A two-dimensional double quantum filter (DQF) experiment is used [17, 18]. This
technique gives a means of isolating the data of interest, i.e. the NMR signal from
the elastomer which is interacting with the carbon black particles, while the signal
from the elastomer not interacting with the carbon black, i.e. the bulk elastomer,
is eliminated. This is accomplished by the DQF by filtering out the signal from the
mobile elastomer protons not interacting with the carbon black. In this way, the
complex solid-state NMR spectra are simplified by observing only the relatively few
elastomer protons that are dipolarly coupled in the immediate vicinity of the carbon
black particles.
In this study we show that the three elastomer components mentioned above can
be identified and analyzed quantitatively using the two-dimensional spectra that are
obtained from DQF experiments on rubber samples.
81
Figure 3-1: Structural properties of carbon black particles [20].
3.2.1
Properties of rubber-like samples
The mechanical properties of the elastomers in rubber samples are enhanced when a
suitable filler is added to them. In most cases the filler used is CB. Fig. 3-1 shows
details about the structure of the carbon black (CB) particles [20]. Carbon blacks
are very fine powders. They consist of aggregates made from partially fused spherical
particles. One primary carbon black particle is made of concentric surface parallel
ordered layers whose graphitic order is diminishing near the particle center [21]. The
diameters of such particles vary from 10-100nm. The surface of CB particles presents
ordered zones of small graphitic crystallites joined by less ordered zones on which
functional groups are located. It can be considered that this is an intermediate state
between amorphous carbon and the crystalline structure of graphite. The presence of
carbon black into a rubber is of significant commercial importance since carbon black
not only enhances the mechanical properties of the final products but also decreases
their cost.
82
!"# Figure 3-2: Polybutadiene (PBD) and the characteristic 1 H dipolar couplings (ωD =
µ0 γ 2
h̄ ) in both cis and trans configurations.
4π r 3
The second important component of rubber samples consists of elastomers, i.e.
polymers responsible of their elastic properties. The polymer used in this study was
low vinyl polybutadiene (PBD) because of its relative simple 1 H NMR spectrum.
The characteristic dipolar interactions between 1 H atoms in PBD have strengths
between 4-21kHz. There can be different dipolar couplings between the same atoms
in polymers as, for example, in the case of cis and trans isomers (see Figure 3-2).
Fast, liquid like intrachain motions average this interaction. However, in our system,
due to the constraints imposed by the carbon black particles, the residual value is not
zero because some degree of motion anisotropy is introduced. The magnitude of the
1
H-1 H dipolar coupling affects the dipolar lineshape in NMR spectra and it contains
information about the local conformation of the elastomers. At the same time, the
NMR signal is also influenced by the intensities of the local fields which arise due
to differences in the local susceptibility induced by the presence of the carbon black
particles. In this study, NMR is used to investigate the physical properties of the
elastomers by aiming to decouple the dipolar and susceptibility interactions.
83
3.3
Method
Our new two-dimensional DQF method is based on the following considerations regarding the specific aspects of filler-elastomer interactions. In the immediate vicinity
of the filler particles, the lines of the external magnetic field are perturbed by their
presence. Therefore, local field gradients appear in their immediate neighborhood.
This effect, due to local susceptibility differences, is reflected in the dynamics of the
nuclear spins from the elastic polymer chains and can be detected by NMR. On the
other hand, the presence of filler introduces strong restrictions on the polymer mobility. Therefore, dipolar interactions that are averaged out by molecular motion in pure
polymer mixtures, can be detected. The dipolar interactions provide information on
the local chain motions while the susceptibility shift provides a mean to estimate the
distance to the filler (CB) and the local geometry. The correlation between the dipolar
and susceptibility interactions provides information on the chain orientation. The 1 H
NMR signal from the rubber samples is due to the protons in the elastomeric chains.
The vast majority of these protons are in elastomer chains that are not interacting at
all with the carbon black and, therefore, they are of no interest in this study. However,
they mask the desired information regarding the carbon black-elastomer interactions.
This is true even in bound rubber fractions where a great deal of molecular mobility
still exists. Our method using two-dimensional NMR spectra investigates the correlations that exist between the NMR signals due to dipolar interactions and signals
that contain both the dipolar and susceptibility influences. This method permits the
characterization of the spatial distribution of the elastomer relative to the carbon
black surface, the mobility of the elastomer, the local order and lengthscales and the
relative amounts of elastomer present in the neighborhood of carbon black particles.
3.3.1
Powder average
The NMR frequencies are dictated by molecular parameters such as segmental orientation. This dependency is essential in understanding and analyzing the twodimensional spectra that we obtain. The spectral lineshape depends on the relative
84
θ
θ
ω
ωχ
Figure 3-3: The NMR frequency depends on the relative molecular orientation with
respect to the external magnetic field (ω = ωD(χ) (1−3 cos2 θ)). By integrating over all
possible orientations, the dipolar and respective susceptibility lineshapes are obtained.
orientation of the external field with respect to the principal axis system (PAS) of the
dipolar and susceptibility tensors [19]. In Fig. 3-3 it is shown the signal calculated
by integrating over all the possible orientations with respect to the external magnetic
field B0 , of a dipole and, respectively, of a local susceptibility field (normal to the
surface). ωD and ωχ reflect the amplitudes of the local dipolar and susceptibility
magnetic fields.
3.3.2
DQF dipolar/susceptibility spectroscopy
In solid state NMR, the couplings between various types of nuclei can be very strong.
This effect leads to very broad NMR spectral lines that are difficult or impossible
to analyze. The common approach to overcome this difficulty is to spin the sample
at very high spinning rates around a direction that makes a “magic angle” with the
85
π/2
π
!"
# # $%!&'
()
'
& *
{
{
{
π/2 π/2
Figure 3-4: Double quantum filtering pulse sequence.
external filed. In this method, known as the Magic Angle Sample Spinning (MASS)
[23, 24] many interactions suffered by the nuclear spins are averaged out and the
spectral lines become sharper allowing a qualitative and, in many cases, a quantitative
characterization of the sample. For a sample dissolved in solution, this averaging is
done naturally, even without MASS, producing very sharp spectral lines in liquid
state NMR. The MASS is very useful in obtaining chemistry information about the
sample, however, other structural information like coupling information is lost by this
averaging. In our study, we make use of two types of interactions: susceptibility and
dipolar. These interactions are averaged out by MASS and, therefore, we perform
the experiments under static conditions.
We use a two-dimensional correlation spectroscopy pulse sequence with double
quantum filtering [25]. The pulse sequence is shown in Fig. 3-4. This sequence
ensures that on the first dimension, during the evolution time t1 , the spins can be
considered as evolving only due to dipolar interactions. The π pulse refocuses the
effect of the time-independent susceptibility fields. The filter ensures that the final
signal is only due to spins that are dipolarly coupled during t1 , strongly attenuating
the signal from the bulk elastomer which would otherwise mask the signals of interest
in our system. On the second dimension (i.e. during the acquisition time t2 ), the
spins evolve under both dipolar and susceptibility fields. In this way, we can study
the correlation between the dipolar and susceptibility influences while the undesirable
signal is strongly attenuated. The detected signal, S(t1 , t2 ), contains the dipole-dipole
interactions from the elastomer’s protons that have greatly restricted mobility, in the
vicinity of the carbon black particles. Also contributing to the remaining signal is the
86
"!
#%$'&)(+*-,/.0*
DFEHG
012436587:9<;>=
('*-,/.0*
012436587I=
?A@
&)(+*-,/.0*
01B4365C;>=
Figure 3-5: Typical 2-D DQF spectral correlation map. The three distinct spectral
regions used in our analysis are emphasized (dotted lines).
additional broadening due to the susceptibility effect. Double Fourier transformation
of S(t1 , t2 ) leads to the two-dimensional NMR-DQF spectrum S(ω1 , ω2 ). The twodimensional spectra obtained with this pulse sequence represent the strength of the
remaining dipolar interaction along one axis and a combination of the dipolar and
susceptibility interactions along the other axis.
3.3.3
Components of two-dimensional DQF spectra
A typical two-dimensional spectrum is shown in Fig. 3-5. In our model we identify
three regions. As expected, in this type of two-dimensional DQF spectral map there is
a component which is symmetric, along the diagonal. This peak is due to the dipolardipolar correlations. We refer to it as the 1st component. Another spectral region
consists of the off-diagonal peaks and it reflects the dipolar-susceptibility correlations.
We refer to it as being the 2nd component. Finally, the 3rd identified component is
the horizontal peak which is due solely to susceptibility influences.
If the spins during both evolution (t1 ) and detection (t2 ) times evolve only under
87
the influence of dipolar interactions, the diagonal cross peaks are obtained. Such
a polymer component, which does not feel any susceptibility influences, might be
located in small cavities of the carbon black particles. There are no susceptibility
variations across these cavities and the motion of the polymer is restricted by the
presence of the carbon black particles in its close vicinity. Therefore we refer to this
component as encapsulated.
The off diagonal peaks are obtained when the signal which during t1 suffers only
dipolar interactions is affected during t2 by both dipolar and susceptibility interactions. The angle between these off diagonal peaks varies with different relative
strength of the dipolar and susceptibility fields. This angle is measured and then
simulations are performed in order to extract the relative strength of the dipolar and
susceptibility fields which are characteristic for this 2nd component. The presence of
the carbon black particles induces local magnetic field gradients in their close vicinity
due to susceptibility differences between carbon black and polymer. This 2nd component in the two-dimensional data is due to polymer which is restricted in motion
by the carbon black but it can also feel the effect of susceptibility differences. This
elastomer component is found on the surface of the carbon black particles (surface
component).
The 3rd component is analyzed and extracted from the projections of the twodimensional signal on the two axes. The spectra obtained are fitted and information
about the local susceptibility field and the amount of molecules responsible for this
component is extracted. The results identify this component as being located further
away from the carbon black particles but still close enough to feel the effect of the
susceptibility fields. We refer to this component as entangled.
To extract quantitative information from the two-dimensional DQF experiments,
we complement the two-dimensional NMR-DQF measurements with theoretical simulations. These theoretical simulations support our model and permit the extraction
of quantitative information about our samples.
88
Figure 3-6: Simulation results showing the two-dimensional spectrum obtained for
different orientations of the PAS between the dipolar and susceptibility tensors. The
simulation with 0o between the PAS of the dipolar and susceptibility tensors is in a
very good agreement with the experimental data.
3.3.4
Orientation of PAS between dipolar and susceptibility
tensors
Using theoretical simulations we can study the relative orientation of the principal
axis system (PAS) between the dipolar and susceptibility tensors. The results of these
simulations are shown in Fig. 3-6. By comparing these theoretical simulations with
the experimental results, we infer that these two PAS are almost parallel for all of our
samples (e.g. see Results, Fig. 3-16). The simulation with 0o between the PAS of the
dipolar and susceptibility tensors is in a very good agreement with the experimental
data.
3.3.5
Simulations of the encapsulated and surface elastomers
We use simulations of the two-dimensional NMR-DQF spectra to quantify the relative
amounts of molecules that are responsible for the 1st and 2nd component. We simulate
the case where the amounts of molecules that feel only dipolar interactions (i.e. the
89
Ι2
Ι1
χ Figure 3-7: Simulation results showing the 1st (encapsulated) and 2nd (surface) elastomer components (see text for details). The maximum intensity peaks that define
the 1st (I1 ) and 2nd (I2 ) component are shown.
1st component) is equal to the amount of molecules responsible for the 2nd component
(Fig. 3-7). The ratio of the maximum intensity peaks that define these components is
1.45. We use these results to analyze the experimental data and to derive the relative
amounts of molecules responsible for these two components.
3.3.6
Relative strength of the dipolar and susceptibility local
fields
The relative contribution of the local dipolar ωD and susceptibility ωχ fields is obtained by monitoring the angle θ between the most intense peaks for various relative
strength of the dipolar and susceptibility fields. Simulation results are shown in Fig.
3-8 for 0.1 <
ωD
ωχ
< 10. The variation of θ with
90
ωD
ωχ
is shown in Fig. 3-8. This result
ω ωχ
θ χ
ω ωχ
χ
#(
θ &)*
"
(
θ &#
χ
ω ωχ
χ
ω ωχ
θ !
θ χ
ω ωχ
χ
ω ωχ
θ ω ωχ
θ ω ωχ
χ
θ &'"(
χ
ω ωχ
# $%
)
θ (#+*+
χ
Figure 3-8: Simulations results showing the variation of the angle θ between the most
intense peaks corresponding to the 2nd component for different relative strength of
the dipolar and susceptibility fields.
will be used to characterize the 2nd component quantitatively.
3.3.7
Lengthscale estimation for the surface component
In order to extract the lengthscale information we performed SEM/TEM imaging
experiments in order to estimate the distributions of the radii of carbon black particles
in our samples. We use this information in numerical simulations with the goal of
characterizing better the encapsulated elastomer component and, at the same time,
of estimating the width of the surface elastomer component.
The field near the surface of the carbon black particle is not uniform. For a
91
300
250
200
150
θ(
100
50
0
0
1
2
3
4
ω
5
ωχ
6
7
Figure 3-9: The variation of θ with for
8
9
10
ωD
.
ωχ
homogeneous sphere of susceptibility χ, the susceptibility field in points outside the
sphere, in polar coordinates, is given by [26]
B(r, θ) = B0 ((1 +
a3 µint − µext
2a3 µint − µext
)
cos
θ
·
i
−
(1
−
) sin θ · iθ ) (3.1)
r
r3 µint + 2µext
r3 µint + 2µext
where a is the radius of the carbon black particle. The value that we use in our
calculations is the value that we determined from the SEM/TEM measurements. µint
and µext are the permeabilities of the carbon black particle and, respectively, of the
surrounding medium; B0 is the external applied static magnetic field. The relation
between the permeability and the magnetic susceptibility is
µ = µ0 (1 + χ)
(3.2)
Knowing the susceptibility difference between the carbon black particles and their
surrounding medium (elastomer), we can estimate the local field around the carbon
black particles. The susceptibility of the carbon black particles can be estimated from
92
experimental measurements (see Results).
Based on these estimations, we can calculate the magnetic field distribution in
spherical shells of various widths around the carbon black particle (Fig.
3-10).
Because the further we go from the surface the field is more homogeneous, the distribution gets more and more narrow. From these simulations, we correlate the 2 nd
moments (M2 ) of these field distributions to the various corresponding widths of each
shell and we obtain the “calibration” curve shown in Fig. 3-11. When the M2 values
are extracted from experimental data, the curve in Fig. 3-11 permits the immediate
estimation of the approximative thickness of surface elastomer component.
3.4
3.4.1
Results and discussion
Two-dimensional DQF measurements on rubber samples
The rubber samples used in this study are shown in Table 3.1. The samples were
provided by Goodyear. For sample selection, emphasis was placed on establishing
a strong foundation for the understanding of the elastomer-filler interaction as opposed to a comprehensive study for different types of elastomers. Initially, low vinyl
polybutadienes were studied (sample code 5G, X92795, 192714G) because of their
relative simple 1 H spectra in order to simplify the interpretation of the data. Later,
blends and functionalized polymers were studied (sample code 4G, capped, 2-arm,
1992720G, 45-1). Unless otherwise stated, the experiments were run on bound rubber fractions (after the toluene-soluble portion of the elastomer was extracted; this
process effectively increases the carbon black concentration by a significant amount
which depends on how much rubber was extracted) in order to eliminate as much as
possible the non-interacting elastomer. Carbon black 779 (CB 779) was used exclusively to eliminate the carbon black type as a factor. This carbon black is a N229 ISAF
reinforcing black with high surface activity (124cc/100g) and surface area (108m 2 /g).
93
2000
% % 1500
% % 1000
% % 500
0
% %
% 500
1000
#
$
2000
B0
1500
1000
500
0
500
1000
2000
1500
1000
500
0
500
1000
2000
1500
1000
500
!" 0
500
1000
Figure 3-10: Simulation results showing the field distribution in spherical shells of
different thickness around the carbon black particle.
94
'
&
√
$%
!#"
Figure 3-11: The dependency of the 2nd moment of the field distribution as a function
of the shell thickness.
sample
Sn-Br
4G
5G
capped
2-arm
X92795
192714G
192720G
45-1
Table 3.1: Sample properties.
filler level (phr)
sample description
70
brf - 0% coupled
79
brf - capped at one end with Sn(Bu)3
213
brf - not functionalized
160
brf - capped at one end with Sn(Bu)3
160
brf - two arms radiating from Sn
40
not brf , not functionalized
96
brf - not functionalized
173
brf - functionalized - SBR
has hexamethylene imine (HMI)
on one end and Sn(t − Bu)3
on the other end
80/20 SBR/PI
40
not brf, functionalized - SBR
has hexamethylene imine (HMI)
on one end but no functional
group on the other end
polymer
LVPBD
LVPBD
LVPBD
LVPBD
LVPBD
LVPBD
80/20 SBR/PI
80/20 SBR/PI
a
a
LVPBD is a low vinyl polybutadiene. A commonly-used term is Li-PBD, where the Li- indicates
that the PBD was initiated with n-BuLi. The microstructure is: 51 trans, 39 cis, 10 vinyl. Sn-Br is a
tin-coupled LVPBD. It is 0% coupled which means that it is same as not tin-coupled, i.e., linear Li-PBD.
SBR is styrene-butadiene. The microstructure is 15 styrene, 60 trans, 11 cis, 14 vinyl. PI is Natsyn
2200, a synthetic polyisoprene (97% cis). brf = bound rubber fraction.
95
"!
Figure 3-12: Two-dimensional DQF spectrum obtained from the sample 5G. Only
the encapsulated (1st component) is present in this sample.
The experiments were done on a Bruker DRX500 spectrometer at a 1 H frequency
of 500.13MHz. The samples were packed into 4mm Zirconia rotors and experiments
were performed using a MAS solid state probe. The experiments were static, nonspinning. The 900 pulse length was 5µs and recycle delays of 3s were used. The t1 time
was varied from 3µs to 3.2ms in increments of 25µs. All measurements were carried
out at room temperature (20 0 C). As explained in the method section, the DQF
method permits the direct measurement of two-dimensional dipolar versus dipolar
and susceptibility correlation spectra.
In Fig. 3-12 are shown two examples of two-dimensional correlation spectra obtained from our samples. We analyzed several types of samples and we observed
important similarities. Two extreme cases are shown in Fig. 3-12. In these figures,
the vertical axis, F1 , corresponds to t1 during which we only have dipolar interactions
while horizontal axis, F2 , corresponds to t2 where both the dipolar and susceptibility
fields influence the spin dynamics.
Fig. 3-12 shows the results obtained on the 5G sample, in which almost the
entire content of free elastomer was extracted. Fig. 3-13 shows the results obtained
on the SnBr sample which has a smaller carbon black concentration. Both samples
initially contained 40 phr CB. The bound rubber fractions (brf) levels are 19% for
96
Figure 3-13: Two-dimensional DQF spectrum obtained from the sample SnBr. All
the three elastomer components are present.
5G and 51% for SnBr and the resulting carbon black levels are 213 phr (5G) and
79 phr (SnBr). There are noticeable differences between these two spectra. For
the 5G sample, we see that there is only one spectral region, along the diagonal.
This signal is due to polymer molecules that are dipolar coupled and do not see any
susceptibility field (1st component). This pattern is identical with the simulation
pattern for a two-dimensional DQF spectrum where there is only dipolar interaction
and no susceptibility effects. It is known from other studies that without carbon black
there are no dipolarly coupled spins. All these are consistent with the picture that
some of the elastomer is encapsulated by carbon black. The projections along both
axes for the 5G sample (the dipolar lineshape) are the identical, as expected, and the
spectral broadening is due to the very limited elastomer mobility (Fig. 3-14). The
signal in the 5G sample is, therefore, from well-structured, highly ordered elastomer
with greatly restricted mobility surrounded by carbon black. This could result from
elastomer filling in crevices or even holes in the carbon black particles or agglomerates
(these possibilities are discussed later in correlation with the results obtained from
scanning electron microscopy measurements). The elastomer in this environment is
intimately in contact with the carbon black and very rigid.
The SnBr data displays all the three elastomer components. Besides the diagonal
97
symmetric peaks which are due to the dipolar-dipolar correlations, we can also distinguish off diagonal peaks which are due to the signal which, during the acquisition
time is affected, besides the dipolar interactions, by susceptibility. These off-diagonal
peaks reflect the dipolar-susceptibility correlations (2nd component). There is also a
third horizontal peak which is due to the susceptibility influences (3rd component).
This component is due to signal which is not evolving during the acquisition time
when the spins feel only dipolar interactions, e.g. the spins are not dipolarly coupled but they feel a strong susceptibility field. The projection on both axes contain
contributions from both dipolar and susceptibility affected spins (Fig. 3-15).
For the SnBr sample and all the other ones, except 5G, there is a 2nd component
and we can characterize its main features quantitatively. We derived that in our
samples the ratio of the dipolar to susceptibility strength is ωD /ωχ = 0.7. Also,
for this 2nd component there is still a significant dipolar coupling (> 2kHz). These
observations are consistent with a near surface material, where there is still a relatively
strong dipolar coupling and also strong susceptibility influences.
The 3rd spectral region is characterized by weak dipolar strength but strong susceptibility local fields. We can distinguish and extract quantitatively two dipolar
components from the F 1 projection from all the samples, except, of course, 5G. Fig.
3-17 shows this for the SnBr sample. The broad dipolar component is due to the
encapsulated and surface elastomer components discussed above and it has similar
lineshape with the F 1 projection from the 5G sample. The narrow dipolar component
is due to the 3rd elastomer component. Significant asymmetry observed in the F 2
projection confirms the very large susceptibility effect. If we think again about the
carbon black being responsible for the dipolar interactions as well as for the presence
of local susceptibility fields, the values for the dipolar and susceptibility couplings for
the 3rd component are consistent with the situation when the polymer is in the area of
large carbon black agglomerates, but far from the surface. Some of this elastomer is
entangled with the elastomer chains near the carbon black surface (2nd component).
In all the samples where the surface component is present, there is also the entangled
component (3rd component).
98
0/1!#"#$'&'() !+*
2/3
-.
!#"%$'&'() !+*
,
-.
Figure 3-14: Two-dimensional DQF spectrum from the sample 5G and the F1 and
F2 projections. The projections along both axes are the identical and the spectral
broadening is due to the very limited elastomer mobility
99
!
,+-*.0/01323456.87
"
#
$
%
&
'(%
)
'$
'#
'"
,+-*.0/132945 .87
"
#
$
%
&
'*%
'$
'#
'"
Figure 3-15: Two-dimensional DQF spectrum from the sample SnBr and the F1
and F2 projections. The broad dipolar component is due to the encapsulated and
surface elastomer components discussed above and it has similar lineshape with the
F 1 projection from the 5G sample. The narrow dipolar component is due to the 3rd
elastomer component. Significant asymmetry observed in the F 2 projection.
100
As shown earlier (see Fig. 3-6, Method), by comparing the general features of the
two-dimensional NMR-DQF spectra with results from numerical simulations, we infer
that for our samples the PAS of the dipolar and susceptibility tensors are parallel to
each other.
3.4.2
Relative strength of the dipolar and susceptibility local
fields
Simulation results of the variation of the angle θ between the most intense peaks in
the 2nd component versus
ωD
ωχ
is shown in Fig. 3-9. If we measure this angle from
the experimental results, it is about 132o for all the samples used in this study (Fig.
3-16). Therefore, for all samples, except 5G, there is a 2nd component and we can
characterize its main features quantitatively. We derived that in our samples the ratio
of the dipolar to susceptibility strength is
3.4.3
ωD
ωχ
= 0.7.
Relative amounts of elastomer components
As we mentioned earlier (see Methods), we can quantify the relative amounts of
molecules that are responsible for the 1st and 2nd component and the simulation
results are shown in Fig. 3-7. The ratio of the maximum intensity peaks that define
these components is 1.45. We use these results to analyze the experimental data and
to derive the relative amounts of molecules responsible for these two components.
Table 3.2 shows the ratio of the encapsulated (1st component) to surface molecules
(2nd component) for all of our samples.
In order to analyze quantitatively the 3rd component, we study the F1 and F2 projections. From the F1 projection we can estimate the lineshape of the 3rd component
by using a Lorentzian fit (Fig. 3-17). After extracting it from the F 1 projection we
are left with the expected dipolar lineshape. The line broadening the the Lorentzian
fit gives us and estimation for the amplitude of the dipolar strength ωD for the 3rd
component. The extracted 3rd component, the Lorentzian, gives the relative amounts
of polymer responsible for this 3rd component. Fig. 3-18 shows the fitting results for
101
!
63
64
"
3
2
*
"
#%$'&(&!)
BDCFEEHGI
1
1
*
/ A
"
"
#-$'&JK)
O :=<?> OP+,
O :=<?>=<
*
#@$'&/.0)
"
*
)
+ 8QL O
#-$'&JK)
,
61
62
5
64
63
7
4
9;:=<?>:8
3
2
#-$'&/.0)
<?L C NM
4
5
62
8 ,
61
+,
/ 7
"
#@$'&(&!)
/ A
"
#@$'&(R!)
Figure 3-16: Two-dimensional DQF results for all the samples used in this study.
The angle θ between the most intense peaks corresponding to the 2nd component is
shown and it is about 1320 for all the samples.
102
Table 3.2: Experimental results: intensity ratio encapsulated/surface
sample Iencapsulated /Isurf ace
Sn-Br
4.54
4G
2.54
5G
capped
2.78
2-arm
3.37
X92795
2.84
192714G
192720G
2.16
45-1
2.34
"!"#%$'&)(
*
$,+-./102+3!
Figure 3-17: F1 projection of the SnBr sample. The 3rd component is fitted with a
Lorentzian lineshape and extracted.
the SnBr sample. The F2 projection can be fitted with an axially asymmetric powder
pattern with large susceptibility strength.
The experimental results are summarized in Table 3.3. This table shows the
relative amounts of encapsulated, surface and entangled with surface elastomer components. The higher the CB level, the higher the relative amount of encapsulated
component. The surface and entangled components are either present together or not
present at all. The unextracted 45-1 sample has the largest relative amount of entangled component and the smallest amount of encapsulated component. This suggests
that some of the entangled elastomer may be extracted out with toluene (assuming equal efficiency in filtering out the mobile component). A direct comparison of
experimental results before and after extraction should be made to confirm this.
103
!"0(21,3&45*6,7(+8:9<;7=>*5%/?
$
!"#
%/@(),
$
&%')(+*+,-.%/
Figure 3-18: F2 projection of the SnBr sample. The 3rd component is fitted with an
axially asymmetric powder pattern.
sample
Sn-Br
4G
5G
capped
2-arm
X92795
192714G
192720G
45-1
M2 (Hz)
F1 proj.
3437
3074
3341
3137
3231
3081
2780
2380
2880
Table 3.3: Experimental results
M2 (Hz) M2 (Hz)
% enc Ienc /Isurf
rd
extr. 3
extr. dip. + surf.
478
3059
83
4.54
592
3239
82
2.54
100
592
3216
89
2.78
592
3322
87
3.37
478
2665
88
2.84
550
3575
98
552
3031
87
2.16
592
2986
58
2.34
104
θ
133
132
134
134
132
133
130
%
ent.
17
18
0
11
13
12
13
42
%
surf.
15
23
0
24
20
23
28
17
%
enc.
68
59
100
65
67
65
59
41
Figure 3-19: AFM image of the 5G sample.
3.4.4
Estimation of CB particle sizes: SEM/TEM/AFM experiments
In order to characterize the three elastomer components that can be identified by
the two-dimensional NMR-DQF, a more detailed study of the carbon black particles
was necessary. Therefore, we employed atomic force microscopy (AFM) [27, 20] and
scanning electron microscopy (SEM) [28, 29, 30, 20] imaging experiments to investigate the carbon black particles in our samples. For AFM imaging, we used two basic
methods: topographical and phase imaging.
Fig. 3-19 shows the AFM images obtained on the 5G rubber sample. On the left
is shown the topographic image, and on the right is presented the phase image. We
observe that the carbon black is relatively uniformly distributed in the sample but it
is quite hard to extract quantitative measurements form this figure. We estimate the
average diameter of the carbon black particles to be about 54nm
We performed measurements on a rubber sample with a lower carbon black concentration (the SnBr sample), and, as expected, we observed that the carbon black
particles are more dispersed. As illustrated in Fig. 3-20, the quality or the AFM
105
Figure 3-20: AFM image of the SnBr sample. As expected, we observed that the
carbon black particles are less concentrated than in the 5G sample.
images of rubber samples permits only a very limited quantitative analysis. This is
due to the problems that are commonly encountered with AFM imaging of rubber
samples, the most important being the sample’s surface preparation. The samples
must be cryomicrotomed and special care must be taken such that the surface does
not oxidize. Typically microtomed sections show signs of surface oxidation within 3060 minutes of preparation. Oxidation and component bloom (surface precipitation of
soluble ingredients from the compound) are two of the biggest hurdles to overcome
when imaging rubber compounds, and they might be a source of the tip contamination. We have, therefore, employed also an alternative imaging method: scanning
electron microscopy (SEM).
Fig. 3-21 shows a typical SEM image obtained for the carbon black powder,
without elastomer. We use this result to estimate the size distribution of the carbon
black particles before mixing them with the elastomer. The average size of the carbon
black particles is 35±4.5 nm. Of course, it would be interesting to see if better
measurements can be performed at higher magnification. Unfortunately, this is not
possible because of artifacts that appear due to the sample coating, charging and
106
Figure 3-21: SEM image of the carbon black powder. The mean diameter of the
carbon black particles is 35±4.5 nm.
sample movement and these effects can be seen in Fig. 3-22
Fig. 3-23 presents SEM results for the SnBr sample which has a relatively small
concentration of carbon black. The best measurements were again obtained from an
intermediate magnification scale. The average diameter in this case was 110±17 nm.
Fig. 3-24 presents SEM results for the 5G sample which has the highest concentration
of carbon black particles. The average diameter in this case was 83±9.7 nm.
An even better resolution however, when possible, is obtained with transmission
electron microscopy (TEM) [31, 20]. However, these experiments are the very difficult
due to sample preparation requirements and can only be performed in a limited
number of cases (Fig. 3-25). The samples must be extremely thin (< 100nm) and,
therefore, we have only used this method for obtaining images of the CB powder.
The results from the SEM measurements are summarized in Table 3.4. These
results permit estimations of size distributions of carbon black particles and can be
correlated with the corresponding observations from NMR measurements regarding
the relative amounts of each elastomer component. By analyzing the SEM/TEM
images it is possible to extract the carbon black diameters. In the case of pure
carbon black powders the mean diameter is significantly smaller than the average
107
Figure 3-22: SEM image of the carbon black powder at higher magnification. Artifacts
due to the sample charging become visible.
Figure 3-23: SEM image of the SnBr sample. The mean diameter of the carbon black
particles is 110±17 nm.
108
Figure 3-24: SEM image of the 5G sample. The mean diameter of the carbon black
particles is 83±9.7 nm.
Table 3.4: Estimations of mean diameters of CB particles derived from SEM experiments
sample
CB779
SnBr
X92795
192720G
5G
mean diam.
(nm)
37.4±4.13
31.17±4.21
29.65±4.75
36.92±4.84
102.56±20.85
155.15
65.67±6.67
72.89±5.71
68.42±10.03
41.87±4.6
56.52±20.11
47.55±6.2
76.6±7.5
94.86±14.0
sample mean
diam. (nm)
34.9±4.5
111.3±17.4
68.8±8.6
48.0±9.7
82.9±9.7
109
# particles
52
19
31
51
5
1
5
6
19
13
10
10
17
9
% ent.
% surf.
% enc.
17
15
68
12
23
65
13
28
59
0
0
100
Figure 3-25: TEM image of the carbon black powder.
110
!#"%$
'&(") *+-,./0"1$
32 *4/0"%,."1$65 73/98
Figure 3-26: Histograms showing the carbon black particle size distributions. The
raw data and the corresponding Gaussian distributions are shown.
carbon black diameter in the rubber samples, when it is mixed with elastomer. This is
an important observation that we are using in the theoretical simulation step in order
to estimate the location of the encapsulated elastomer. These measurements suggest
that some elastomer might enter inside the structure of the carbon black particles.
At the same time, the results from our two-dimensional NMR-DQF measurements
which can identify the three elastomer components, show that we have systematically
a larger fraction of encapsulated elastomer in cases when larger carbon black particles
are observed. For example, for sample 192720G, whose main difference from the others
is that the elastomer is functionalized and contains larger aromatic rings of styrene
we measure smaller diameters than for the other rubber samples, yet still larger than
for pure carbon black. This suggests that it is more difficult for the elastomer to enter
the carbon black particles.
Fig. 3-26 shows the distribution of carbon black particle sizes extracted from the
SEM/TEM images. Both the raw data and the corresponding Gaussian distributions
are shown. It is apparent that the average carbon black sizes are quite different in
various types of rubbers. It seems that the presence of the elastomer is correlated
with higher carbon black diameters.
111
3.4.5
NMR susceptibility measurements of carbon black
It is possible to measure the susceptibility difference between the carbon black and
another reference substance in a MASS NMR experiment. As a reference we used
(dimethylsulfoxide) DMSO. The NMR spectrum of DMSO consists of two narrow
lines, 0.3 ppm apart. The spectrum of DMSO mixed with carbon black powder shows
besides the two narrow lines, two broad lines which are due to the DMSO trapped
between the CB particles (Fig. 3-27). The broad peak is due to the susceptibility
effects. In a static external field, the carbon black particle induces a magnetic dipolar
field around it. This field is not homogeneous and is responsible for the broadening of
the DMSO peak. The carbon black susceptibility can be measured from the distance
between the narrow and broad peaks [22]. The splitting between these two peaks is
196 Hz for the CB779 in a 500MHz spectrometer. The observed signal splitting is
given by [22]
2
∆ν = ν0 ∆χ
3
(3.3)
where ν0 = γB0 is the main static magnetic field and ∆χ is the susceptibility difference
between carbon black and DMSO. According to Eq. 3.3 and the measured splitting
between the two peaks, we can calculate the susceptibility difference between carbon
black and DMSO, ∆χ = 6.15 · 10−6 .
3.4.6
Characterization of the encapsulated elastomer
Another question that we can address using numerical simulations is regarding the location of the encapsulated elastomer component. From the results of two-dimensional
NMR measurements we see that this component does not feel the effects of susceptibility fields. A first hypothesis is that these elastomers could be trapped between
aggregates of carbon black particles. As we have seen from the SEM/TEM experiments, it is very common to have carbon black particles agglomerates rather than
single, isolated particles. Therefore, this situation seems plausible. To test this hypothesis, we estimate the field distribution in a small region between the carbon black
particles. We start by using two particles. As shown in Fig. 3-28, for two spheres,
112
CB + DMSO (field = 600MHz; νr = 5kHz)
1
CB779 ∆ν = 196Hz
CB630 ∆ν = 289Hz
CB889 ∆ν = 192Hz
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5.5
5
4.5
frequency (ppm)
4
3.5
Figure 3-27: Carbon black susceptibility measurements. Three different types of CB
were analyzed. The red curve corresponds to the CB779 which was used in or study.
113
Figure 3-28: Simulation results for two spheres showing the field distribution in a
small region between the carbon black spheres.
large field susceptibility variations occur at these locations for various orientations
with respect to the external magnetic field B0 . Because the volume investigated is
relatively small the field distributions are quite narrow. However, there is a broad
range of field values if we consider the entire sample with all the possible orientations and, therefore, the space between two spheres is not a possible location for the
encapsulated elastomer.
A more realistic case would be to consider the space inside clusters of three and
even five spheres, tangent to each other. For three spheres the simulated field distribution is still quite broad (see Fig. 3-29). Even for five spheres we observe that
the simulated distributions are still very broad, only slightly narrower than for the
case of three spheres (see Fig. 3-30).
All these simulation results suggest that the
encapsulated component, since it does not feel susceptibility variations, is probably
114
θ=0
θ = π/2
Figure 3-29: Simulation results for three spheres showing the field distribution in a
small region between the carbon black spheres.
θ=0
!"
θ = π/2
Figure 3-30: Simulation results for five spheres showing the field distribution in a
small region between the carbon black spheres.
115
not trapped between the carbon black agglomerates. It must be found in a different
location. This observation coupled with the information from the SEM/TEM measurements, suggests that the encapsulated elastomer is rather trapped between the
shells of the carbon black particles themselves inducing the increase of their effective
radius as observed by SEM. The currently accepted model for carbon black particles
which was proposed by Heidenreich R.D. et al. [21] by using TEM allows for the
possibility of a specific carbon black/elastomer interaction in which the elastomers
disrupt the ordered semicrystalline structure of the graphitic ordered plates at the
surface of carbon black particles producing an increase of their effective volume. As
shown in Ref. [21], the average spacing measured by electron microscopy, between
the CB graphitic layers, is about 0.4 nm. This could allow the elastomer chains to fit
between the graphitic plates.
3.4.7
Lengthscale estimation for the surface component
The SEM/TEM imaging experiments produce reasonable estimations for the distributions of the radii of carbon black particles in our samples. We use this data, as
described above, to find the thickness of the surface component. Since the susceptibility difference between the carbon black particles and their surrounding medium
(elastomer) is known, we can estimate the local field around the carbon black particles and calculate the magnetic field distribution in spherical shells of various widths
around the CB particle. The variation of the 2nd moments of these field distributions
with the widths of each shell is shown in Fig. 3-11. By comparing the 2nd moments of
these distributions with corresponding values from the two-dimensional experiments
(shown in Fig. 3-18), we can estimate the width of the elastomer component on the
surface of the carbon black. Results of estimations using data from the NMR experiment suggest a width of about 1-3nm. Fig. 3-31 shows a model of the surface and
entangled components relative to the CB surface.
116
Figure 3-31: Schematic model showing the surface (red) and entangled (green) elastomer components in the vicinity of the CB particle.
3.5
Conclusions
We present a new method based on the analysis of two-dimensional NMR-DQF correlation spectra for the study of heterogeneous samples. The method is applied to
the characterization of the two main components in rubber samples (i.e., the elastomer and the filler particles) and their interactions. Our method, combines the
results of direct, experimental measurements with theoretical simulations, to allow
for a quantitative estimation of the relative amounts of elastomer components and
their classification in relation to their proximity to the carbon black surface. Three
elastomer components are identified in our samples: encapsulated, surface and entangled with the surface component. It was observed that in all samples there is a set of
molecules that are dipolarly coupled but that do not see any susceptibility field. Since
in the absence of CB there are no dipolarly coupled spins, we infer that these elastomer molecules must be encapsulated in the CB environment. For this component,
the calculations showed that even in small interior regions inside clusters of carbon
black particles there are important susceptibility field variations. This observation,
together with the SEM/TEM results which consistently showed a “swelling” of the
carbon black particles in the presence of elastomer, suggests that the encapsulated
component is located between the graphitic plates rather than between carbon black
particles.
117
By comparing the experimental data with theoretical simulations we found that
the PAS of the dipolar and susceptibility tensors corresponding to the 2nd component
are parallel to each other. This suggests that the polymer at the surface of CB particles is ordered. By analyzing the 2nd moments of the F1 projections, we conclude that
the motion of encapsulated and surface polymers have similar magnitudes. We found
that the relative amplitudes of the dipolar and susceptibility fields corresponding to
the 2nd component is
ωD
ωχ
= 0.7. For the elastomer at the surface, our method permits
the estimation of a width of 1-3nm.
We have also estimated that the strength of the local dipolar field ωD for the 3rd
component is small. This corresponds to a polymer component entangled with the
surface component, which is quite mobile. At the same time, this 3rd component is
also under the influence of a strong susceptibility local field.
Our new method offers a direct way to obtain detailed quantitative information
on the elastomer components in rubber samples and can contribute to a better understanding of their specific structural and physical properties. Further questions
that need to be addressed for fully understanding the complex properties of rubber
samples are concerned with a description of the elastomer in the immediate vicinity
of the CB surface (first 1nm from the CB surface) and with the investigation of the
ordered surface elastomer component.
118
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[13] V.M. Litvinov, P.A.M. Steeman, Macromolecules, 32, 8476 (1999).
119
[14] R. Mansencal, B. Haidar, A. Vidal, L. Delmotte, J.M. Chezeau, Polym. Int., 50,
387 (2001).
[15] H. Lüchow, E. Breier, W. Gronski, Rubber Chem. Technol., 70, 747 (1997).
[16] V.J. McBrierty, K.J. Packer, Nuclear Magnetic Resonance in Solid Polymers,
Cambridge University Press (1993).
[17] G. Bodenhausen, R.L. Vold, R.R. Vold, J. Magn. Reson., 37, 93 (1980).
[18] G. Jaccard, S. Wimperis, G. Bodenhausen, J. Chem. Phys., 85, 6282 (1986).
[19] K. Schmidt-Rohr, H.W. Spiess, Multidimensional Solid-State NMR and Polymers, Academic Press, London (1994).
[20] J.B. Donnet, R.C. Bansal, M-J. Wang, Carbon Black: science and technology,
M. Dekker, New York, USA (1993).
[21] R.D. Heidenreich, W.M. Hess, L.L. Ban, J. App. Cryst., 1, 1 (1968).
[22] J-H. Chen, B.M. Enloe, D.G. Cory, S. Singer, unpublished results.
[23] E.R. Andrew, A. Bradbury, R.G. Eades, Nature, 182 1659 (1958).
[24] I. Lowe, Phys. Rev. Lett., 2, 285 (1959).
[25] R.R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press, Inc., Oxford (1994).
[26] J.D. Jackson, Classical Electrodynamics, John Wiley and Sons, Inc. (1999).
[27] G. Binning, C.F. Quate, C. Gerber, Phys. Rev. Lett., 56, 930 (1986).
[28] A.V. Crewe, J. Wall, Proc. of Electron Microscopy Society of America, 27th
Annual Meeting, St. Paul, Minnesota, p. 172 (1969).
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120
[30] I.L.F. Ray, I.W. Drummond, J.R. Banbury, Developments in Electron Microscopy
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121
122
Chapter 4
NMR identification of fluids and
wettability in preserved cores in
situ
4.1
Summary
The determination of the chemical composition and wetting fluid of the mobile phase
in core samples are essential elements in understanding fluid dynamics in these systems. A combination of NMR, magic angle sample spinning (MASS) and relaxation
measurements permits this determination in a simple, efficient and unambiguous fashion. MASS removes the effects of variation in the bulk magnetic susceptibility that
would otherwise degrade the spectral information and prevent measurement of the
chemical composition. Relaxation measurements permit the components in contact
with the surface to be identified since the surface relaxation dominates all other contributions to relaxation. We demonstrate this by a series of measurements on two
preserved cores taken from a sandstone formation.1
1
The material presented in this section is published in Petrophysics 43(1) 13-19 (2002).
123
4.2
Introduction
In this chapter we describe high-resolution nuclear-magnetic-resonance (NMR) experiments on fluids contained in two preserved core samples to identify the types of
fluids, and, to determine the wettability state of the rock, without cleaning or modifying the native state of the rock. In both samples of rock studied here the water and
oil peaks are clearly resolved by magic angle sample spinning (MASS), permitting a
direct measurement of their relaxation rates. This was not possible without MASS.
Chemical information such as fluid type and relative saturations and wettability
are vital information needed for reservoir evaluation and management. Proton, 1 H,
NMR-spectroscopy has emerged as a rapid, non-destructive measurement, both in the
laboratory and in the oil fields. In the logging industry, the relaxation of the proton
magnetization is widely used to infer petrophysical properties such as pore size, fluid
viscosity, etc. [1, 2]. The in situ differentiation of various chemical species, however,
is not possible with current NMR logging tools. Even in laboratory measurements,
the estimation of oil and water content in a rock sample involves time-consuming
extraction and distillation processes [3].
In simple fluids, the proton chemical shift δ is easily used to identify the chemical
composition. Since this shift is proportional to the external field, B0 , a high magnetic
field is desirable. On the other hand, the susceptibility broadening is proportional
to the field strength. Therefore, in systems with large variations in the susceptibility
(such as rocks), higher magnetic fields do not lead to enhanced spectral resolution.
Thus, while the high-resolution NMR spectroscopy is routinely used as a standard
analytical tool, the application of high-resolution techniques to such complex systems
as fluid-saturated rocks has been hindered by the spectral line broadening due to
susceptibility contrasts between the constituents.
MASS removes the orientationally dependent susceptibility broadening [4] while
leaving the orientationally independent part of the chemical shift. This is achieved by
spinning the sample rapidly about the [111] direction with respect to the static field.
This is a rotation about the body diagonal of a cube which systematically exchanged
124
the 3 cartesian axes and thus reduces dipolar interactions to their isotropic average
(or zero). Recently, MASS has been used to study oil and water in rocks [5, 6, 7, 8].
In our analysis, we use MASS to suppress the susceptibility broadening and then
combine this with T1 and T2 (Carr-Purcell-Meiboom-Gill (CPMG) or Hahn Echo
(HE)) measurements [9].
Wettability is revealed by a comparison of different relaxation rates. Wettability
is an important factor which determines the residual oil saturation, capillary pressure,
and relative permeability. The importance of wettability in determining displacement
of oil by water and residual oil saturation is well established. It is of great interest
to design experiments to characterize wettability that are less time consuming than
core-flooding and imbibition experiments.
The spin-lattice relaxation rate, T1 , of bulk water is about 3 seconds and in rocks
it is substantially reduced. This is attributed to the paramagnetic impurities on the
pore wall [10] which give rise to a surface relaxivity ρ. When the pore size, a is
sufficiently small compared to a relaxation length, Lρ = D/ρ, relaxation is in the
fast-diffusion limit (where D is the diffusion coefficient of the fluid) [11]. In this
instance, the longitudinal and the transverse relaxations are given by:
1
ρS
≈
,
T
Vp
(4.1)
where S/Vp is the surface-to-volume ratio of the pore space in a rock. This relation
(Eq. 4.1) holds for an isolated pore. It is rather crude for systems with well connected
pores, such as rocks, nevertheless, it is widely used in the logging industry (see [1, 2,
6, 12] and references there in).
The above model can be used to understand, in a qualitative manner, the effect
of wettability. First, consider a oil wet rock. If a layer of oil molecules on the wall
prevents the water molecules from interacting with the paramagnetic impurities at
the pore wall, the water relaxation rate should be close to its bulk value. In this case
the oil is expected to decay faster than its bulk value. However, in a water wet rock,
a layer of water will protect the oil molecules from the wall and water will decay fast.
125
For example, Straley [2] found that the T2 of kerosene in a rock was greatly reduced
from its bulk value if the rock had been dried carefully before saturating it with
kerosene. This is because kerosene molecules could come in contact with the pore
wall in a well-dried sample. If, however, the drying was insufficient before saturating
the rock and a thin film of water remained, the kerosene relaxed at its own bulk rate.
The layer of water molecules protected the kerosene from the impurities on the wall.
The MASS experiment affords the opportunity of measuring the relaxation of oil and
water simultaneously and thus of determining the wetting fluid.
One might worry that the centrifugal forces under MASS would distort the fluid
distribution in the rock. However, the fluids hardly move during spinning. This can
be seen from the Bond number or the Etövös number [13] which measures the ratio
of centrifugal force to the surface tension
Eo =
ρω 2 ra2
<< 1
σ
(4.2)
Here ρ is the density of the fluid, ω is rotational frequency, r is the distance of the
fluid element from the axis of the rotation, σ is the fluid surface tension and a is the
pore size. These parameters may have quite a wide range of values as they depend on
the type of fluid, wettability, pore-size etc. Using typical values of σ ≈ 20 dynes/cm,
a ≈ 100µm, r ≈ 2mm and a rotor frequency of 8kHz, we find that surface tension
dominates and flow is negligible.
4.3
X-ray data
The two preserved cores used in these experiments are mainly silicates. The main
minerals present in the two samples are listed in Table 4.1.
X-ray micro scans were taken using a SCANCO µCT80 tabletop X-ray microscanner. Images taken at 60 kVp are shown below in Figures 4-1 and 4-2. The grains
of sample A are larger, on average, than the grains in sample B. X-ray absorption
measurements at 60, 50, and 40 kVp indicate that the larger grains, in both samples,
126
Table 4.1: Quantitative X-Ray Diffraction Data Analysis (Weight Percent)
Sample A
Sample B
QZ K-SPAR ALB SID
94
02
01
00
85
05
03
01
KAOL ILL SM ANA
TR
TR 01
TR
01
01
01
TR
PYR AMP HM
TR
TR
TR
TR
TR
00
a
a
QZ = Quartz; K-SPAR = Potassium Feldspar; ALB = Albite; SID = Siderite; KAOL = Kaolinite;
ILL = Illite; SM = Smectite; ANA = Anatase; PYR = Pyrite; AMP = Amphibole; HM = Hematite.
Figure 4-1: CT image of Sample A. The sample holder, seen as the black outline, is
2 cm in diameter.
127
Figure 4-2: CT image of Sample B. The sample holder, seen as the black outline, is
2 cm in diameter.
are quartz. The white speckles are due to highly absorptive, dense minerals. Although present only in trace amounts (< 5% by weight), the combination of Anatase,
Pyrite, Amphibole, and Hematite may be sufficient to produce the observed white
speckles. The quantitative X-ray analysis shown in Table 4.1 was done separately in
a mineralogy laboratory.
4.4
MASS data on rocks
The static spectra of the two samples taken at two different fields, 12T and 3T are
broad and featureless, see Figures 4-3 and 4-4.
Under MASS, the spectrum for each species splits into a central peak and a series
of sidebands. We spun the sample sufficiently fast such that the sidebands do not
overlap with the center line of water and with the three center oil peaks. All the
spectra were taken at 200 C and at two different fields: 3T, using a Bruker AMX
spectrometer and 12T, using a Bruker DRX spectrometer, each equipped with a
128
high-resolution 1 H/13 C MASS probe with gradient coils. The sample was packed in
a 4-mm MASS Zirconia rotor. The spectra are shown in Figures 4-3 and 4-4. The
oil has at least three peaks – alkanes to the right of water and aromatics to the left
of water. However, in the spectra taken at the lower field, even when spinning, we
can not distinguish the three oil peaks. This is due to the short T2 values (i.e., the
intrinsic linewidth which is not due to the susceptibility difference). Therefore, we do
our investigation only at the higher field.
"#%$'&)(*,+
021!354!6
"#%$'&)(-*/.
78:9
021!354!6
ωB-C
78;9
EDFHG
!
!
!
<!=>=?A@
!
Figure 4-3: UPPER PANELS: MASS taken at 3T while the sample is spun at 8KHz.
LOWER PANELS: respective static spectra.
4.5
Inversion-recovery with MASS
The spin-lattice relaxation time was measured by the inversion-recovery method.
With the sample spinning, a hard 1800 pulse of duration 8.8µs inverts all the spins
129
!
"$#&%(')+*
!
"$#&%(')?>
,-/.10
,-/.
,-=.=<
46587:9;
46587:9;
,-=.=<
,-=.32
46587:9;
46579@;
,-=.=<
DCEGF
ω;BA
,-/.=<
,-=.10
,-=.
,-/.32
H 757 -/I
0
Figure 4-4: UPPER PANELS: MASS taken at 12T while the sample is spun at 8KHz.
Oil has, at least, three peaks — alkanes to the right of water and aromatics to the
left of water. LOWER PANELS: respective static spectra.
130
which are then allowed to relax (at the spin-lattice relaxation rate) for a “recovery”
period tR . A final 900 pulse excites the magnetization in the transverse plane after
which the signal is collected. The Fourier transform of the signal as a function of
tR is shown in Figure 4-5. The data are fitted using a three-parameter, non-linear
procedure to obtain the longitudinal relaxation time, T1 , listed in Table 4.2.
"!$#&%('
"!$#)%+*
./1054
./1054
./1032
./1032
6879;:<
6879;:<
,-
Figure 4-5: MASS-Inversion Recovery on Sample A and Sample B at νr =8KHz These
results give for Sample A: T1 (water)=2.1s, T1 (oil1)=0.51s, T1 (oil2)=0.51s and for
Sample B: T1 (water)=1.04s, T1 (oil1)=0.56s, T1 (oil2)=0.58s. Note that T1 of water
in Sample A is close to its bulk value, while in Sample B it is approximately half its
bulk value.
4.6
CPMG with MASS
The spin-spin relaxation rate was measured via the Carr-Purcell-Meiboom-Gill method.
With the sample spinning and following an initial 90o excitation pulse, a series of 180o
pulses are applied synchronized with the rotor frequency. In addition to surface relaxation, the spins in the transverse plane relax by destructive interference in the
magnetization, as each spin sees a different magnetic field and precesses at a slightly
different rate. There is also relaxation by diffusion in the inhomogeneous fields. The
series of echoes are collected and the Fourier transforms for the two samples are shown
in Figure 4-6 as a function of the echo time.
131
A bulk sample of dead oil from the same reservoir has the following characteristics
at ambient Temperature and Pressure:
T2 = 267.12 ms
Viscosity = 1.445 centipoise
API gravity = 30.8 at 60o F
Molecular Weight = 229 g/mol; main component = CH4 (53.21% mole fraction)
with other alkanes and higher components.
*,+.-0/2143?>
*,+.-0/214365
; =< ; '
< 798 : ; =<
; =<
798@: "!$#%'&(&()
"!$#%'&(&()
Figure 4-6: MASS-CPMG on Sample A and Sample B (νr =8KHz, τe =125µs).
We applied a numerical fit known as the Uniform PENalty (UPEN) to fit multiexponentials [12] to the Inversion-Recovery and CPMG data to obtain the T1 and T2
distributions. Figures 4-7, 4-8 and 4-9 show these distributions for the water and the
two oil peaks. As we have mentioned before, the mean T1 of the largest fraction of
the water molecules in sample A is longer than the mean T1 of the largest fraction of
the water molecules in sample B. In addition, the mean T1 for sample A is closer to
the mean T1 of bulk water, while for sample B it is much shorter. Therefore, we infer
that for sample A the water is in poor contact with the pore walls, while in sample
B the water relaxes mainly at the pore surface. Thus, the sample A is not water wet
while sample B is mixed wet.
The T2 data of water in Figure 4-7 shows that, in sample A the distribution of
pore sizes is peaked for relatively large pores, which corresponds to large values of T 2 .
132
This agrees qualitatively with the X-ray pictures, Figures 4-1 and 4-2. In sample B
the T2 distribution of the water is broader which implies that there is a more uniform
distribution of pore sizes. We note that sample B has a smaller number of very large
pores than the sample A has, as suggested by Figures 4-1 and 4-2.
The T1 and T2 distributions for the oil peaks look very similar to each other (see
Figures 4-8 and 4-9). The value of T1 , where the peak of the T1 distribution for water
occurs, is smaller for sample A than for the corresponding T1 for sample B. This is
consistent with the assumption that sample A is not water wet and sample B is mixed
wet. We also see that the largest fraction of oil molecules in sample A has a T2 value
smaller than that in sample B. This suggests that in sample A a greater number
of large pores contain oil. For sample B the T2 distribution looks more uniform,
consistent with our conclusion that sample B is mixed wet.
!#"
!#"
!%$
!%$
&
'
&
'
Figure 4-7: T1 and T2 distributions for the water peak in both Sample A and Sample
B.
4.7
Conclusions
The results are summarized in Tables 4.2 and 4.3. At least three oil peaks are present
in both samples. Presumably some of the lower alkanes were lost by evaporation when
the core was exposed to the atmosphere. The water fraction of sample A is greater
133
"!
!
"
!
"
!
"
#%$
# $
%
# &
'
# &
'
(
(
)
)
Figure 4-8: T1 and T2 distributions for the oil1 peak in both Sample A and Sample
B.
*
?
>A@=CB"D
? - A> @=CB"D
? 0 >A@=CB"D
? - >A@=CB"D
0
+4 <
+4 ;
E%F
E F
%
E G
H
E G
H
? 5
+4 :
I
J
? *
K
J
? *
0
K
+4 9
+4 8
+4 7
+4 6
+4 5
+4 *
+
*+,-
*+/.
*+-
=>
*+0
*+21
*+3
Figure 4-9: T1 and T2 distributions for the oil2 peak in both Sample A and Sample
B.
134
Table 4.2: Longitudinal Relaxation Times
Sample
Sample A
Sample B
Species
water
oil1
oil2
water
oil1
oil2
T1 (s)
2.10
0.51
0.51
1.04
0.56
0.58
Table 4.3: Chemical Shift
Species δ (ppm) δ (Hz)
water
4.8
2400
oil1
1.8
900
oil2
1.3
650
oil3
7.1
3550
than that of sample B, and the value of its longitudinal relaxation time, T 1 ≈ 2.1 sec
in sample A, is close to the relaxation time of bulk water, T1 ≈ 3 sec, which suggests
that water in sample A is not in contact with the rock. Sample B has T1 ≈ 1.04 sec,
which implies that the water in this case relaxes by contact with the rock. The T 1
of the major oil peaks are about 0.5 sec in both samples suggesting that the nature
of oil-grain contact, if any, is about the same for both the rocks. We can therefore
assert that if sample A is not water wet and it is only oil wet, then sample B is mixed
wet. Conventional wettability measurements were not done on these samples, but
both the samples were believed to be mixed wet.
Surface-induced NMR relaxation can indicate the wettability state. Further work
is needed to study the influence of wettability by simultaneous observation of relaxation of oil and water in well-characterized water-wet, oil-wet, and mixed-wet systems.
NMR experiments are less time consuming than core-flooding and imbibition experiments. This procedure may eventually be developed as a quick and non-destructive
measure of wettability.
135
Additional work is required to establish the relationship between NMR measures
of wettability and the macroscopic measures of wettability characteristics which are
used in the petroleum industry, e.g., contact angles, capillary pressure, relative permeability, or resistivity of liquid-saturated rocks.
136
Bibliography
[1] W.E. Kenyon, Nuclear Magnetic Resonance as a Petrophysical Measurement,
Nucl. Geophysics, 6, 153 (1992).
[2] C. Straley, G.E. Morriss, W.E. Kenyon, J.J. Howard, The Log Analyst, p. 36,
Jan-Feb (1995).
[3] J.W. Amyx, D.M. Bass, R.L. Whiting, Petroleum reservoir Engineering” McGraw Hill, New York (1988).
[4] A.N. Garroway, J. Magn. Reson., 49, 168 (1982).
[5] T.M. de Swiet, M. Tomaselli, M.D. Hürlimann, A. Pines, In Situ NMR analysis
of fluids contained in sedimentary rocks, J. Magn. Reson., 133, 385 (1998).
[6] D.M. Wilson, G.A. LaTorrca, NMR magic angle sample spinning for measuring
water and heavy oil saturation and high resolution relaxometry, SCA paper 9923,
Soc. of Core Analyst (1999).
[7] G. Leu, X-W Tang, S. Peled, W. Maas, S. Singer, D.G. Cory, P.N. Sen, Amplitude Modulation and Relaxation due to Diffusion in NMR Experiments with a
Rotating Sample, Chem. Phys. Lett., 332:(3-4), 344 (2000).
[8] Y. Liu, G. Leu, S. Singer, D.G. Cory, P.N. Sen, Phase and Amplitude Modulation of Spin Magnetization in Magic Angle Spinning NMR in the Presence of
Molecular Diffusion, J. Chem. Phys., 114, 13 (2001).
[9] E.L. Hahn, Phys. Rev., 80, 580 (1950); H.Y. Carr, E.M. Purcell, Phys. Rev., 94,
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137
[10] F. Bloch, Phys. Rev., 83, 1062 (1951).
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388 (1961).; J.D. Robinson, J.D. Loren, Soc. Pet. Eng. J., 249, 268 (1970).
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Press (1978).
138
Chapter 5
General conclusions
In this work are presented new approaches and methods for the study of soft and
porous materials. Examples of interesting materials include reservoir rocks and synthetic polymer with fillers. For these, many physical properties such as pore size
distribution, pore geometry, interaction types between different components (e.g. wettability in rocks, elastomer-carbon black interaction in rubber), are not completely
understood and additional characterization methods are needed.
MASS NMR is a widely used method for studying solid state materials. In our
work we extended the MASS method to include the effects of molecular diffusion,
which is inherent in heterogeneous media that contain also a more mobile, fluid-like
phase. This new set of methods, named “diffusive MASS NMR” is presented in the
first part of the thesis (Chapters 1 and 2, Ref. [1, 2, 3, 4]) and summarized in Ref.
[5]. Molecular diffusion leads to a disruption of the signal refocusing accomplished
by MASS and introduces an amplitude modulation of the rotary echoes. The new
diffusive MASS methods are designed to separate the amplitude modulated signal
from the isotropic spectrum (which reports on the chemistry of the mobile phase)
and from the frequency modulation (which is due to the static field variation and,
thus, to the local structure of the solid phase). Diffusive-MASS takes advantage of the
high resolution spectra obtainable with MASS (which allows the simultaneous identification of various chemical species present in the sample) and, at the same time,
of the probing of the local gradients by molecular diffusion. The main advantage
139
of the new NMR method is that it offers a non-invasive, direct mean to probe the
chemical composition and the dynamics of the mobile phase through a porous, heterogeneous material. The results for the first method on investigating the amplitude
modulation and relaxation in diffusive MASS NMR presented in Chapter 1 (Ref. [1])
were obtained primarily by the author of this thesis. The following experiments (Ref.
[2, 3, 4]) were performed in collaboration with Yun Liu. This work was developed in
collaboration with Schlumberger-Doll Research.
A second type of NMR methods for characterizing heterogeneous samples is presented in Chapter 3. These methods [6] were developed with the goal of being directly
applicable to studies of filled elastomers. The presence of filler particles (such as carbon black) induces local susceptibility fields while the local ordering of the elastomer
introduces dipolar fields. The separation of susceptibility and dipolar influences allows
for the direct observation of the elastomer components as reflected by the elastomercarbon black interactions. The NMR method has several advantages over alternative
methods used in polymer studies, such as light, X-ray or neutron scattering, electron
microscopy, dielectric or mechanical relaxation. Due to the missing periodicity in
these systems X-ray and neutron scattering provide limited information. Scattering
techniques have focused the physical description of solid polymers on the crystalline
parts, while the amorphous parts were often treated only as filling material. Multidimensional NMR permits different spin interactions to be correlated or separated.
Our method permits the direct probing of sample properties such as: chain mobility,
thickness of the elastomer on the surface of the rigid phase, and relative amounts of
elastomer components, which are not directly observable by other techniques. This
work was performed in collaboration with Goodyear Research. While the main motivation was to develop a mean to characterize the specific elastomer-filler interactions
in rubber samples, the method developed here is more general being applicable to the
study of other heterogeneous samples in which the interaction between a mobile and
a rigid phase is present.
In the last part, Chapter 4 are described high-resolution MASS NMR studies on
fluids contained in preserved core samples. This study [7] allows the identification
140
of the types of fluids, the determination of relative pore size distributions and the
characterization of the wettability state of rock samples without modifying its native
state. While the relaxation of proton magnetization is widely used to infer petrophysical properties in the logging industry, the differentiation of various chemical
species is not possible with current NMR well-logging instruments. The MASS NMR
method allows a simple and efficient characterization of the chemical composition
and wetting fluid in rock samples. These measurements can be performed faster than
core-flooding and imbibition experiments which are currently employed by the petrophysical industry. These results were obtained in collaboration with ExxonMobil
Exploration Company and Schlumberger-Doll Research.
141
142
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Phys. Lett., 332:(3-4) 344 (2000).
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(2001).
[3] P.N. Sen, Y. Liu, G. Leu, D.G. Cory, Chem. Phys. Lett., 366, 588, (2002).
[4] Y.Liu, G. Leu, S. Singer, P.N. Sen, D.G. Cory, J. Chem. Phys., 119(5), 2663
(2003).
[5] P.N. Sen, Y. Liu, G. Leu, D.G. Cory, Magn. Reson. Imaging, 21(3-4), 257 (2003).
[6] G. Leu, Y. Liu, D. Werstler, D.G. Cory, in preparation.
[7] G. Leu, A. G. Guzman-Garcia, D.G. Cory, P.N. Sen, Petrophysics, 43(1), 13
(2002).
143