Diffusion Ordered Nuclear Magnetic Resonance Spectroscopy: Principles and.
Applications. ... Also, from lecture notes of Dr. Péter Sándor of Varian Inc.
NMR Spectroscopy: Principles and Applications Nagarajan Murali Exchange, Diffusion Lecture 11
Dynamics We talked about motions introducing a coupling between spins and its surrounding (lattice) leading to relaxation. Such motions are said to be in a time scale comparable to the reciprocal of Larmor frequency time scale. There are other types of motions in different time scales that affect NMR spectrum uniquely and such process can be delineated from the spectrum. The ability to extract motional information from the analysis of NMR spectrum renders this technique its uniqueness and this subject is known as dynamics by NMR.
Motional Time Scale Time scales of typical motional mechanisms relevant in NMR is summarized below:
Exchange, Diffusion In this lecture, we will focus on chemical exchange observed by NMR and effects of molecular diffusion. Exchange Spin Dynamics – Malcolm H. Levitt, John Wiley & Sons Diffusion High Resolution NMR Techniques in Organic Chemistry – T.D.W. Claridge, Chapter 9. Diffusion Ordered Nuclear Magnetic Resonance Spectroscopy: Principles and Applications. C.S. Johnson et al., Progress in NMR Spectroscopy 34 (1999) 203-256. Suppression of Convection Artifacts in Stimulated Echo Diffusion Experiments. Double Stimulated Echo Experiments. A. Jerschow and N. Mueller, J. Magnetic Resonance 12, (1997) 372-375. Also, from lecture notes of Dr. Péter Sándor of Varian Inc.
Exchange If the motional processes involve making and breaking of chemical bonds and or conformational changes, then such processes are called Chemical Exchange.
Translational Motion In a liquid, molecules can undergo translations. Translations that are random and uncoordinated is called diffusion. Translational motion that is concerted and directed is called Flow.
Exchange NMR is capable of detecting Chemical Exchange of reactions in equilibrium and also detect exchange between indistinguishable reactants and products.
Exchange Let us focus on the symmetrical, slow (spectral time scale) two site exchange. A
k k
B
The forward and backward reaction rate constants are equal (k s-1). The probability of making a transition in an interval t is kt. Five representative molecules jumping between the states A and B all starting from state A. k=3x10-3 s-1.
Exchange Let us say the chemical shift of a resonance when the molecule in state A is WA and it is WB in state B. Also define W = WA - WB , then the exchange is termed slow are fast based on the relative magnitude of k with respect to W .
k W 2 Slow Exchange k W 2 Fast Excahnge k W 2 Crossover point
Exchange Let us now see what happens to the magnetization MA when the molecule is state A with resonance frequency WA and MB in state B with frequency WB . W tI
z M A (0) cos W t M A (0) sin W t M xA (t ) A x A y A
W tI
z M A (0) cos W t M A (0) sin W t M yA (t ) A y A x A
M A (t ) M xA (t ) iM yA (t ) [ M xA (0) iM yA (0)] cos W At [ M xA (0) iM yA (0)]sin W At M A (t )
M A (0)eiW At
M B (t ) M B (0)eiW Bt
d d M A (t ) M A (0)e iW At dt dt iW A M A (0)e iW At iW A M A (t )
d B M (t ) iW B M B (t ) dt
Exchange The rate of change MA and MB including the T2 relaxation and exchange is then: M A (t ) d A A M (t ) iW A M (t ) kM A (t ) kM B (t ) dt T2 B M (t ) d B M (t ) iW B M B (t ) kM B (t ) kM A (t ) dt T2
A d M (t ) iW A R k dt M B (t ) k
A M (t ) iW B R k M B (t )
k
The solution of this equation is M A (t ) M A (0) exp( Lt ) M B (t ) M B (0) iW A R k L k
k
iW B R k
Exchange The observed signal can be arrived at as: 1 ik 1 ik S (t ) (1 ) exp( d1t ) (1 ) exp( d 2t ) 2 2 1 d1 (W A W B ) i k R 2 1 d 2 ( W A W B ) i k R 2
W A W B 22 k 2
R
1 T2
if k W A W B 2
iR
i T2
if k W A W B 2
Exchange The observed signal will then appear as: Slow exchange
Fast exchange
Exchange The lineshape has information on the motion time scale:
Asymmetric Exchange In case the forward rate and reverse rate are different as below, A
k K’
B
Then the equilibrium constant K=k/k’ is equal to the ratio of the equilibrium concentrations [B]eq/[A]eq. In the fast exchange limit, k>>|(WA-WB)|/2, we get a single peak with position given by the mean of the two chemical shifts weighted by equilibrium concentration.
Asymmetric Exchange From the rapid exchange spectrum one could get the equilibrium rate constant. A
k K’
B
W peak
[ A]eq W A [ B]eq W B [ A]eq [ B]eq
W A KW B 1 K
Exchange – Longitudinal Magnetization The rate of change and including the T1 relaxation and exchange can also be calculated: I Az (t ) d I Az (t ) k I Az (t ) k I Bz (t ) dt T1 I Bz (t ) d I Bz (t ) k I Bz (t ) k dt T1 1 k k I ( t ) T1 d Az 1 dt I Bz (t ) k k T1
I Az (t )
I Az (t ) I (t ) Bz
The z-magnetization exchange can be observed by a NOESY like experiment.
Exchange – Longitudinal Magnetization The 2D spectrum of the pulse sequence below show rate of change and by exchange and the cross-peak and diagonal-peak intensities are given as:
a A B (t ) sinh( kt ) exp{(k T11 )} adiag (t ) cosh( kt ) exp{(k T11 )} sinh( kt )
1 kt 1 e e kt ; cosh( kt ) e kt e kt 2 2
a A B sinh( kt ) tanh( kt ) adiag cosh( kt ) kt for short mixing time t (kt 1)
Exchange – Longitudinal Magnetization The timescale of dynamics probed by the z-magnetization exchange is shown below.
Diffusion Let us now focus on the translations that are random and un-coordinated called diffusion and the principle behind the measurement by NMR.
Diffusion The mean square root of distance a molecule moves in a given time t along z-direction by diffusion is given as 1 z rms (2 Dt ) 2
where D is the diffusion constant. For a spherical molecule of radius r, D
k BT 6 r
Stokes-Einstein Equation
kB is the Boltzmann constant and is the viscosity of the liquid. Larger molecule diffuses slowly.
Diffusion by NMR Diffusion can be measured by NMR by the use of pulse sequences that utilize pulsed field gradients.
Echo amplitude is maximum if =0 or no diffusion and when the area of the two gradients are equal.
( z ) zG
( z ) zG
Diffusion by NMR In the spin echo sequence, diffusion introduces attenuation of the echo signal as molecule randomly moves from one position to another and the net displacement along z during the interval between the gradients. B( z ) Gz
( z ) B( z ) Gz ( z, ) ( z ) Gz
Diffusion by NMR The time evolution of the transverse magnetization M+=Mx+iMy can be written including diffusion as M M i0 M iGzM D 2 M t T2 Free precession
Precession due to gradient
Diffusion
The free precession and T2 relaxation can be transformed away by setting M ( z, t ) exp( i t t / T ) and ( z, t ) (t ) exp iz t g (t ' )dt '
0
2
0
( z, t ) igz ( z, t ) D 2 ( z, t ) t t' t 2 ln (t ) D q (t ' )dt ' where q(t ' ) g (t" )dt" 0 0 2 t t ' (t ) exp D g (t" )dt" dt ' 0 0
Diffusion by NMR The observed intensity of the signal in the gradient spinecho experiment with a rectangular gradient can be derived from the equation for (t) as: exp G 2 D I (G ) I 0 exp 3 T2
I0 is the intensity of echo without any gradients. If and are constants and only the gradient strength is varied the decay due to T2 is also constant in all the experiments and the observed intensity is simply depend on the diffusion decay. 2 I (G ) I 0 exp G D 3
Stejskal-Tanner Formula
Diffusion by NMR
Diffusion by NMR
Diffusion by NMR
Diffusion by NMR
Pulse Sequences for Diffusion by NMR The gradient stimulated echo pulse sequence is the often used experiment to measure diffusion coefficients by NMR. The magnetization is stored along z-direction while the diffusion is operative. PFGSTE (Pulsed Field Gradient STimulated Echo)
PFGLED (PFG Longitudinal Eddy Delay)
t
t
t
t
The longer T1 is operative than the T2
tR
Additional tR delay allows for stronger gradients and eddy current recovery
Bi-Polar Gradient Diffusion Experiments These sequences that use bi-polar (+ and – gradients) are also gentler on the lock and yield clean lineshapes. BPPSTE (Bipolar Pulse Pair STimulated Echo)
t/2
t/2
t/2
/2 /2
t/2
/2 /2
GCSTE (Gradient Compensated STimulated Echo)
t
BPPLED (BPP Longitudinal Eddy Delay)
t/2
t/2
/2 /2
t/2
t/2
tR
/2 /2
GCSTESL (Gradient Compensated STimulated Echo Spin Lock)
t
t
t
tSL
Flow - Convection Fluid flow (concerted motion) in NMR tube is generally an outcome of temperature gradient along the sample and is pronounced in the vertical z- direction.
Flow interferes with the measurement of diffusion as the decay of the signal is accelerated due to the additional translational motion.
Flow - Convection The effect of flow can be compensated under some circumstances, such as flow is laminar and flow is equal in both upward and downward directions.
These conditions are harder to meet with increasing tube diameter or the sample length.
Flow - Convection It can be shown that when the flow is laminar with constant velocity v, the equation of motion for M+ can be solved with the solution: 2 t t ' t t ' (t ) exp D g (t" )dt" dt ' exp iv g (t" )dt"dt ' 0 0 0 0
In reality, the velocity will not be constant along the tube and the superposition of all velocities will reduce the oscillatory flow integral as an attenuation. Also the imaginary part of the integral vanishes when the flow is equal in upward and downward directions along the tube.
Convection Compensation It should be noted that the attenuation by diffusion depends on the integral of the total gradient area square while the phase dispersion from flow depends just on the integral of the total gradient. Thus if the net phase dispersion from the gradients in the entire pulse sequence goes to zero the attenuation due to flow can be removed to first order. t t '
2
t t ' g ( t " ) dt " dt ' 0 g ( t " ) dt " dt ' 0 0 0 0 0
The following pulse sequence satisfy the condition:
Convection compensated diffusion experiments utilizes pulse sequences with gradients and coherence transfer pathways arranged to satisfy such a condition.
Convection Compensated Diffusion NMR
Here the molecules diffuse during delay T that is split into two equal parts and the coherence transfer pathway selection unwinds the phase due to flow during the second T/2 delay that was wound by the first T/2 delay.
Solid Line 0 -1 Dotted Line 0 +1
0
+1 +1
0
-1
0
-1
0
-1
0
+1
0
-1
-1
Convection Compensated Diffusion NMR To verify presence of convection flow, one could miss-set the two parts of the diffusion delay.
No convection
2
-45
0
Convection
+45
temp=25 (T=120 msec, 2 varies from –45 to + 45 msec in 5 msec steps) oC
-22
0
+22 msec
temp=60 (T=60 msec, 2 varies from –22 to + 22 msec in 2 msec steps) oC
Convection Compensated Diffusion NMR Convection is pronounced at higher temperature and for solvents like CDCl3 as shown for the sample here.
H3C O 13CH
3I
+
O
P 13
CH3
CH3
O
Convection Compensated Diffusion NMR Convection is pronounced at higher temperature and for solvents like CDCl3. temperature = 30 0C Normal Sequence
Convection Compensated
1 G/cm
30 G/cm
Convection Compensated Diffusion NMR Convection is pronounced at higher temperature and for solvents like CDCl3.
D (m2/s*10-10)
F1
Normal Sequence
Convection Compensated
