NMR Velocity Mapping of Couette Flow Using Oscillating Magnetic Field Gradients

JOURNAL OF MAGNETIC RESONANCE,

Series A 117, 150–163 (1995)

NMR Velocity Mapping of Couette Flow Using Oscillating Magnetic Field Gradients JEFFREY A. HOPKINS, ROBERT E. SANTINI,

AND

JOHN B. GRUTZNER

Department of Chemistry, Purdue University, West Lafayette, Indiana 47907-1393 Received December 9, 1994; revised June 26, 1995

A method for obtaining spin-density distributions of selected layers in annular shear flow is introduced. A transverse rotating magnetic field gradient is generated by oscillating orthogonal gradients in quadrature. The equivalence of the frames of reference of a rotating sample with a fixed magnetic field gradient and a rotating magnetic field gradient with a fixed sample is demonstrated with modulation sideband patterns. When the magnetic field gradient and sample rotation frequency differ, sidebands appear at integer multiples of the difference between the two rotation frequencies. When the gradient is synchronized with the sample rotation, the spectrum is a spin-density projection of the sample onto the gradient. When the sample and gradient are counterrotating, high audio-frequency sidebands appear with consequent spectral simplification. When the orthogonal components used to generate the rotating gradient are imperfectly matched, sidebands appear at integer multiples of the sum and difference frequencies of Vs and Vg . A novel, low-cost electronic system is described to create the rotating gradient on a standard high-resolution spectrometer. Spectra of layers of fluid moving at a chosen angular frequency in a sample undergoing Couette flow shearing are shown. The radial fluid layer is selected by stroboscopically sampling with a Hahn-echo refocusing pulse applied at integer multiples of the rotation period. A two-dimensional analysis is given that separates the signal arising from the layer that is synchronized with the rotating gradient from the signals arising from the rest of the sample. Complex phase distortions occur when the echo delays do not match the rotation period. A radial frequency map of the Couette streamline flow is generated using the 2D technique which compares favorably with the calculated flow field. q 1995 Academic Press, Inc.

INTRODUCTION

Magnetic-resonance imaging techniques have recently been used to examine rotating bodies, in both solid state and fluid samples (1–6). Images have been made with Cartesian gradients oscillating in quadrature to produce a magnetic field gradient that rotates synchronously with a solid ( 1). Cylindrical radiofrequency gradients have been produced that utilize the symmetry of a rotating cylinder (7). Bodies that undergo rotation relative to a magnetic field gradient experience a modulated magnetic field. In high-resolution 1064-1858/95 $12.00 Copyright q 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.

m4277$0749

12-08-95 01:31:23

NMR, periodicity in the effective magnetic field leads to spinning sidebands (8). In this paper, we extend our method for obtaining spectra of selected fluid layers in sheared Couette flow based on their different rotational frequencies (9). We show that, for a sheared fluid, a given layer can be sampled by stroboscopic application of pulses timed to the rotation period for that slice. The selection was not ideal as the signal from the rest of the sample is not fully selfdestructive (9). Further, the intensity is necessarily T 2 weighted and low-rotation-frequency layers may have periods much longer than T 2 . Rotational Couette flow is an elementary flow in fluid dynamics (10). Stable, two-dimensional flow is generated in fluid contained between two cylinders when the outer cylinder is rotated and the inner cylinder is fixed. A continuum of concentric circular streamlines is obtained with rotational frequencies which range from the cylinder frequency at the outer wall to zero at the inner cylinder (Fig. 1). Alternatively, if the outer cylinder is fixed and the inner cylinder is rotated, complex three-dimensional flow fields—Taylor vortices (11)—result. In this paper, we present images of two-dimensional Couette flow. We previously reported (9) a time-domain treatment of the sideband patterns generated when a cylindrical sample is rotated in a linear, transverse magnetic field gradient. Magnetization created in the transverse plane by a 907 pulse experiences a sinusoidally oscillating magnetic field which causes the precessional frequency of the magnetization to oscillate. The relative phase of the precessional frequency sinusoid is determined by position in the sample and direction of sample rotation relative to the field. The phase acquired by the magnetization from discrete regions is proportional to the area under the precessional frequency curve from that region (Fig. 2). The signal is detected as a free induction decay which is the sum of all the magnetization in the sample. Figure 2 depicts the complicated phase relationship of the magnetization in different regions of the sample. For a uniformly rotating sample, all precessional frequencies vary at the same rate, and the area under all the curves is zero after one complete rotation. The magnetization

150

maga

AP: Mag Res

MAPPING OF COUETTE FLOW USING OSCILLATING GRADIENTS

FIG. 1. Definition of parameters: a and b are the radii of the inner and outer tubes with rotational frequencies V1 and V2 , respectively. Sample position and magnetic field gradient are expressed as vectors in cylindrical coordinates, (r, u ) and (G, f ) respectively, with the angular coordinate chosen with reference to the direction of the x magnetic field gradient component. The origin is at the center of the sample tube and corresponds closely to the virtual zero point of both magnetic field gradient components. When the outer tube is rotated at V2 independently of the inner tube, V1 , the angular velocity of sample layers, Vs , depends on their radial position as shown.

comes back into phase giving a rotational echo. Rotationalecho spectra (Fig. 3a) were detected when the patterns were sampled with a Hahn spin-echo pulse sequence (Fig. 3b). Stroboscopic sampling was achieved by matching the time delay in the echo sequence to an integer multiple of the rotational period (Fig. 3c). Significant phase distortions were observed and simulated when the spin-echo refocusing pulse delay did not correspond to integer multiples of a rotation period (Fig. 3d). However, stroboscopic sampling of fluid layers under Couette flow conditions was less than ideal because of the overlap of phase distorted signals from the ‘‘off-cycle’’ layers. When the magnetic field gradient is synchronized to the rotation frequency of the sample, a train of spin-echo sequences will cause complete refocusing even when arbitrary delays are chosen (Fig. 3e). Figure 4 shows three characteristic spectra of the annular fluid under different flow conditions with a linear magnetic field gradient. Figure 5 shows the distortions that result from the spin-echo pulse sequence. The simulated spin-density projections of the inner (5c) and outer (5e) layers demonstrate the difference in peak-to-peak separations that are measured to determine the radial position of the fluid synchronized with the gradient (5c, static gradient; 5e, rotating gradient). We have now developed a quadrature summation system

m4277$0749

12-08-95 01:31:23

maga

151

to create a rotating magnetic field gradient that can be synchronized with the rotating fluid, in much the same manner that Veeman et al. used to image rotating solids (1). Two AC signals (in quadrature) are imposed on the standard xand y-axis shim coils. The vector sum of these two linear gradients is a rotating gradient. Precision control of the gradient frequency allows synchronization with layers of fluid experiencing shearing in Couette flow. The refocusing pulse(s) may be applied at any time to rephase the magnetization from the layer synchronized with the gradient. Multiple inversion pulse trains (Fig. 3e) may be applied with periods much less than the period of a rotation. This limits signal loss from fluids with relaxation times short compared to the rotation period. Simulation of the magnetization development requires the addition of the rotating gradient vector to the method described previously (9). The treatment can be extended to include the spin echo and to describe signals from a layer of chosen frequency in a Couette flow experiment. A peak-to-peak separation in the frequency-domain pattern from the stroboscopically selected layer is measured to determine the radius for the layer of fluid whose angular

FIG. 2. Rotational echo. The shaded circles represent the time-dependent spatial positions of selected regions of a cylindrical sample rotating in a linear magnetic field gradient (G). The sinusoids represent the timedependent precessional frequency ( v ) of transverse magnetization at two arbitrary radii and supplementary angular coordinates. The small circles show the phase accumulated by transverse magnetization at the outer radius of the shaded portion of the sample. The phase is proportional to the area under each curve. For simplicity, a representative area has been shaded in the top left portion of the diagram only.

AP: Mag Res

152

HOPKINS, SANTINI, AND GRUTZNER

FIG. 3. Pulse sequences. (a) Rotational echo. Echoes generated by a uniformly rotating sample in a linear magnetic field gradient. See the legend to Fig. 2 for a description of the magnetization development. Echoes are formed at every complete revolution ( p/ V, p Å 0, 1, 2,rrr). (b) Hahn-spin echo. The pulse sequence generates an echo at time 2t. The refocusing (1807 ) pulse at time t switches the quadrants of the dephasing magnetization vectors leaving the precessional direction invariant. (c) Stroboscopic sampling. If the spin-echo refocusing pulse is applied at integer multiples of the rotation period (stroboscopically), complete rephasing is accomplished. (d) ‘‘Off-cycle’’ sampling. If the refocusing pulse does not occur at an integer multiple of the rotation period (off-cycle), rephasing is incomplete and amplitude and phase distortions are observed. (e) Multiple refocusing pulses with matched gradient. If the sample is stationary with respect to the gradient, a series of Hahn-spin echoes can be generated without regard for the rotation period.

frequency is matched to the frequency of the magnetic field gradient. THEORY

A general treatment for oscillating magnetic field gradients and rotating samples will be described. A classical approach to derive analytical equations for the time development of the magnetization will be developed. The results will then be reexamined using a different integration scheme which provides additional insight on the subtle effects of experimental parameters. The sample is a concentric tube arrangement with an inner tube of radius a spinning at V1 and an outer tube of radius b spinning at V2; the sample resides in the annulus (Fig. 1). The radial frequency profile of the fluid is described by Vs(r) Å

V2b2 0 V1a2 a2b2 V1 0 V2 / 2 b2 0 a2 r b2 0 a2

[1]

Two experiments were examined initially, uniform spinning, V1 Å V2, and Couette flow, V1 Å 0. For the first case, Vs(r) Å V; for the second,

m4277$0749

12-08-95 01:31:23

maga

Vs(r) Å

V2b2 [r2 0 a2] . r2 [b2 0 a2]

The magnetic field gradients are assumed linear and centered on the sample. They are described as a pair of orthogonal gradient components, Gx and Gy, constrained to oscillate 907 out of phase by the vector generator. The components are given in units of radians per second per millimeter, gB*(b)/b, where B*(b) is the strength of the magnetic field at the outer wall of the tube. The angular frequency of the magnetic field is Vg. The general expression for the magnetic field experienced by a molecule is given by the dot product of the magnetic field vector with the spatial vector, Grr. The phase of the magnetic field oscillation at the time of the initial 907 pulse is given by f Å arctan(Gy /Gx), such that a pure x field has a phase of 07. This gives the time evolution for the oscillating magnetic field as G Å Gxcos(f / Vgt)iO 0 Gysin(f / Vgt)jO ,

[2]

which describes a clockwise sense of rotation in a right-

AP: Mag Res

MAPPING OF COUETTE FLOW USING OSCILLATING GRADIENTS

153

FIG. 4. Calculated characteristic spectral patterns. (a) Spin-density projection of an annular sample onto a linear magnetic field gradient. The sample is stationary with respect to the gradient. The peak-to-peak separation is the product of the magnitude of the magnetic field gradient and the diameter of the inner tube. (b) Spinning sideband pattern generated when the annular sample is uniformly rotated in a linear magnetic field gradient. The sidebands are separated by the frequency of rotation and are referred to by order from the central (isotropic, zero order) line. (c) Sideband spectrum generated when the annular sample is subjected to Couette flow shearing in a linear magnetic field gradient. The pattern is the overlap of sidebands (as in (b)) generated by the continuum of rotation frequencies present in Couette flow (Fig. 1).

FIG. 5. Spectral distortions arising from spin-echo pulse sequences and gradient conditions of Figs. 3c–3e applied to an annular fluid sample rotating in a linear magnetic field gradient. Spectrum (a) was obtained under uniform sample rotation conditions; spectra (b–e) were obtained under conditions of Couette flow shearing. (a) Sideband phase and amplitude distortions resulting from application of sequence 3d, t Å 0.4 V. (b) Spectrum generated by application of sequence 3c, p Å 1. (c) Subspectra of (b) generated by the layer of fluid at the outer wall (sidebands) superimposed on the calculated spin-density projection of the fluid at the inner wall (intensities have been normalized to the volume of the layer of fluid). (d) Spectrum generated by application of sequence 3e, t Å 1/ Vg , t* Å 0. (e) Subspectra of (d) arising from the inner fluid layer (sidebands) superimposed on the calculated spin-density projection of the outer fluid layer (normalized intensities).

handed coordinate system. The resulting magnetic field experienced by a nuclear moment starting at position (r; u) is given by the dot product

can be analyzed separately with no loss of generality by assuming the on-resonance condition in the rotating frame. The development of the magnetization under the influence of the magnetic field gradient is evaluated by

Grr Å Gxcos(f / Vgt)iO 0 Gysin(f / Vgt)jO 1 r cos(u / Vst)iO 0 r sin(u / Vst)jO Å

G/ r cos[u 0 f / (V0)t] 2 /

G0 r cos[u / f / (V/)t] 2

[3]

with G/ Å Gx / Gy, G0 Å Gx / Gy, V0 Å Vs 0 Vg, and V/ Å Vs / Vg. When the strengths of the orthogonal magnetic fields are not matched, an elliptical magnetic field gradient is formed that can be decomposed into counter-rotating components with frequencies V0 and V/ (Fig. 6). At time zero, the bulk longitudinal magnetization is pulsed to form in-phase magnetization in the transverse plane. Static field components, containing the chemical-shift information, can be factored. The influence of the magnetic field gradient

m4277$0749

12-08-95 01:31:23

maga

FIG. 6. Vector depiction of the rotating gradient. If ÉGxÉ x ÉGyÉ, an elliptical rotating magnetic field gradient is formed. The rotating ellipsoid can be decomposed into its Cartesian components (top) or counterrotating radial components (bottom). The magnitude of the component rotating at Vg is G/ Å ÉGx / GyÉ/2. The magnitude of the counterrotating ( 0 Vg ) component is G0 Å ÉGx 0 GyÉ/2.

AP: Mag Res

154

F *

t

M(r, u, t)}exp 0ig

B*(t*)dt*

t0

H

Å exp 0i

G

HOPKINS, SANTINI, AND GRUTZNER

index. The summation over n is no longer independent and is not needed. The summation over k can be factored out to group all k dependent terms together:

J

G/r (sin[u 0 f / V0t] 0 sin[u 0 f]) 2V 0

H

M(t)}

J

G0r 1 exp 0i (sin[u / f / V/t] 0 sin[u / f]) . 2V/

Jj [4]

*∑ a

[5]

nÅ0`

in Eq. [4] gives

1

F GF G

∑ J ( j /m ) 0k k

rG0 rG/ Jk exp[ 0ik2f] 2V/ 2V0 1

b

∑

a j,k ,m ,n

M(t)} Jj

*

b

∑

a

rG/ rG0 Jm Jj /m[W ] 2V0 2V/

1 exp[ 0ij( V0t / 2f )]exp[ 0imV/t]

F

j,m

S

0rG/ sin(2f ) 2V0W

rG/ rG/ rG0 rG0 Jk Jm Jn 2V0 2V0 2V/ 2V/

*

r WÅ

2p

exp[ 0i( j 0 k / m 0 n) u]

0

1

du 2p

2prdr , V

[6]

where V is the volume of the annulus. We begin to gain some insight into the appearance of the spectrum. The only instance where the angular integral does not go to zero is when the combination of indices in the last exponential is zero. The indices must satisfy n Å j 0 k / m, leaving three independent and one dependent index. One of the indices not involved with the frequencies is chosen for the dependent

m4277$0749

12-08-95 01:31:23

DG

1

1 exp[ 0i( 0 j / k / m 0 n) f]

maga

[7]

F GF G

F GF GF GF G

1 exp[ 0ij V0t]exp[ 0imV/t]

1

2pr dr . V

This expression can be further simplified to include only two indices, one for the sum of the frequencies and one for the difference by the substitutions n Å 0k and J0n Å exp(inp )Jn . This allows [7] to be expressed in terms of positive indices only, with the addition of an extra exponential term. The application of Graf ’s theorem of Bessel function addition allows the summation over k to be removed with the introduction of a new argument and phase angle:

1 exp i( j / m)sin 01

Jj

*

1 exp[ 0imV/t]exp[ 0ij( V0t / 2f )]

j,m

`

M(t)}

rG/ rG0 Jm 2V0 2V/

b

If the magnetic field gradients are matched in magnitude so that G0 is zero, the second term on the right is unity. The expression becomes identical to the expression for a rotating sample in a static, linear magnetic field gradient with a new displacement angle (u 0 f) and an apparent rotation rate of V0, the difference between the two rotation rates. In practice, the strengths of the gradients can be roughly matched by minimizing the sidebands that occur at the sum of the two frequencies, V/. The detected signal is proportional to the integral of b 2p Eq. [ 4 ] over the volume of the sample *a rdr *0 du. The integral over r is complicated and is not evaluated analytically. It represents a sum over the annular layers rotating with frequencies given by Eq. [1] . The integral over u is necessary to account for the symmetry of the system and to simplify the expression. Substitution of the Bessel function identity exp[iz sin( x )] Å ∑ exp(inx )Jn (z)

F GF G

S D S D S rG/ 2V0

2

/

rG0 2V/

2

/

D

2pr dr V

r 2G0G/ cos(2f ) . 2V/ V0

[8]

This version appears more complicated, but leaves intact the exponentials which are time dependent in integer multiples of the sum and difference frequencies between the two oscillations. The W terms are complicated phase and intensity factors for the sidebands of order j or m . The outcome is that we observe peaks at the sum and difference of the two rotation frequencies with incoherent phase errors if the pulse does not occur at f Å 0. At this point, it might be convenient to take a different approach to the integral over u. Prior to the integration over

AP: Mag Res

155

MAPPING OF COUETTE FLOW USING OSCILLATING GRADIENTS

time in Eq. [4], the u dependence can be taken outside the integral. The result is collected on terms of sin( u ) and cos( u ) to yield

** b

M(t)}

a

H F S* F S*

2p

0

H

1 exp i sin( u )

** b

Å

a

G/ r 2

exp 0i cos( u ) G/ r 2

t

t0

t

sin( V0t * 0 f )dt *

t0

a

n

m

exp{ 0i zc(r)cos( u )}exp{i zs(r)sin( u )}

0

S D

H*

du p exp[i(n / m) u] 2p 0

J

* ∑ J [zc(r)]J [zs(r)]exp S in p2 D 2pVrdr b

Å

n

a

G0 r 2

/

S*

t

cos( V/t * / f )dt *

t0

t

sin( V/t * / f )dt *

t0

DGJ

DGJ

durdr V [9]

of the rotation frequency appearing in the equation. To simplify the math, we will concentrate on the formation of the simple sideband pattern for uniformly spinning tubes ( V1 Å V2 Å V ) . The analytical expression for the Bessel function argument is

S D S D S D G/r 2V0

2

[2 0 2 cos( V0t)] 2

/

G0r 2V/

/

G/G0r 2 2V0V/

[2 0 2 cos( V/t)]

n

n

(c m Å 0 n).

[10]

In this case, the application of Graf ’s theorem yields a deceptively simple expression

M(t) }

G0 r 2

du rdr . V

Z(r) Å

p 2prdr Jn [zc(r)]Jm[zs(r)]exp 0in 2 V 2

1

D

/

2p

b

D S*

cos( V0t * 0 f )dt *

The terms in square brackets have been designated zc ( r ) and zs ( r ) for the u independent terms associated with cos ( u ) and sin ( u ) , respectively. Using the identity 0cos ( u ) Å sin ( u 0 p / 2 ) and the Bessel function identity [ 5 ] again, this expression can be expanded to [10 ] which, treated as before, yields a sum over a single index:

*∑∑

glean some understanding of how the Bessel function argument of [11] leads to sidebands in the frequencydomain spectrum without the explicit integer multiples

*

q

b

J0 ( zs(r) 2 / zc(r) 2 )

a

2prdr V

b

* J ( Z(r))2pVrdr . q

Å

0

[11]

b

On resonance, there is no imaginary component. This is not obvious in the preceding equations, but is necessary to lead to the symmetric pattern that is required by the symmetry of the system. Since we already know the results from [ 8 ] , we can

m4277$0749

12-08-95 01:31:23

maga

1

5

cos[2( f / Vgt)] 0 cos(2f / V/t) 0 cos(2f 0 V0t) / cos(2f )

6

. [12]

The argument has time dependent terms that are independent of f and oscillate at V 0 and V/ . When Gx equals Gy , G0 is zero, the last two terms drop out, and sidebands occur at integer multiples of V0 . The appearance of terms [1 0 cos ( Vt ) ] under the radical in the argument must give rise to undistorted sidebands at the corresponding frequencies. The last term is not so easy to understand but gives rise to the phase dependence at the corresponding frequencies. There is also a term at the gradient oscillation frequency that was not evident before. We now consider the response of the system to a refocusing pulse. The influence of a spin-echo pulse can be examined semi-classically where the effective accumulated phase angle for a nuclear moment is given by extension of Eq. [ 4 ] :

AP: Mag Res

156

HOPKINS, SANTINI, AND GRUTZNER

* B * (t * )dt * 0* B * (t * )dt * t

M(r, u, t) } exp

t0

ig

2t/ t

. [13]

t

The excitation pulse is applied at time t 0 Å 0; the refocusing pulse is applied at time t; and detection begins at time 2t (Fig. 3b). Again, the results are equivalent to those derived previously with the added complication that when the gradients are mismatched, an extra phase angle is involved relating the positions of the counter rotating gradients. However, when the gradients are matched, we can simplify the Bessel argument in [11] to a manageable length: Z(r) Å

S D G/r 2V0

1

2

[14]

6 / 2 cos[ V0 (t / 2t )] 04 cos[ V0 (t / t )] 04 cos( V0t )

with G Å G/ /2. Treatment as before leads to an expression independent of f and t. As expected, the magnetization refocuses completely regardless of when the pulse is applied:

.

We can recognize the results by comparison to [12]-sidebands arising from terms [1 0 cos( Vt)]. Equation [14] can be decomposed into two separate sets of sidebands at {V0 with different phase dependence and a phase-only term. The right side of Eq. [4] holds if Vs x {Vg . If the frequencies are equal or inverses, one of the exponentials is modified to account for the matched frequencies: Vs Å {Vg

H

c exp 0i

1 exp

Gx { Gy r cos( u | f )t 2 0i

FIG. 7. 31P spectrum (81 MHz) of 50% H3PO4 in D2O in a 10 mm tube with a concentric, 5 mm tube containing H2O. Both tubes are rotating at 20 Hz in a 3.48 mT/m (60 Hz/mm 31P) gradient which is rotating at 083 Hz. The primary sidebands appear at multiples of the difference between the two rates, 103 Hz.

Gx | Gy r 4V

J

sin( u { f / 2Vt) 0 sin( u { f )

[15]

.

When the gradients are matched, the expressions are greatly simplified and some interesting results can be observed. All the standard results observed with a rotating sample and a static gradient (9) are recovered with some added flexibility (8). For instance, when the sample and gradient are counterrotated, very high apparent spinning rates can be achieved (Fig. 7). The matched frequency results need to be evaluated for the spin-echo experiment. In this case, Eq. [13] becomes

F S*

2t/ t

M(r, u, t) } exp 0i

Gr cos( u 0 f )dt *

t

* Gr cos( u 0 f )dt * DG , t

0

[16]

0

m4277$0749

12-08-95 01:31:23

maga

M(t) }

*

b

a

J0[Grt]

2prdr . V

[17]

The Fourier transform of this expression is the familiar frequency projection of an annular sample onto a linear magnetic field gradient (Fig. 4a). These results are general, and the extension to Couette flow is straightforward. The flow profile is independent of u, and the integral over r has not been explicitly evaluated above; therefore, all Vs terms may be replaced by the Vs (r) of Eq. [1]. The spectrum from an annular layer is found by selecting the magnetic field rotation frequency to match the angular frequency of that annular layer in the Couette flow profile. Integration over r sums the signal from all the layers of the sample. Treatment of the integration as a discrete sum allows each layer to be examined separately in simulations. The frequency selected layer will appear to be stationary relative to the gradient while all the others will appear to be in motion (cf. Fig. 5). The spin-echo sequence can be applied to refocus the signal from this ‘‘stationary’’ layer while all the other layers give the complex behavior described above. The time delay ( t ) in the stroboscopic sampling sequence may be incremented to select each layer in the flow profile in turn. Clearly, a sequence of these time experiments can be created and the results Fourier transformed to form a 2D frequency array (12). The F2 frequency axis is the standard spectral axis defined by the acquisition time constants. The F1 axis is defined by the echo delay times. The change in delay is invisible to the stationary layer of fluid. All the other layers produce sidebands that exhibit oscillations with the change in echo delay. This adds a frequency dependence to the signals from the ‘‘off-cycle’’ annular layers. The data

AP: Mag Res

MAPPING OF COUETTE FLOW USING OSCILLATING GRADIENTS

157

output is a 2D array in which the spectrum from the stationary layer is found by selecting the F2 spectrum for which F1 Å 0. EXPERIMENTAL

A vector generator system was developed in the Instrumentation Facility at Purdue University, Department of Chemistry. The generator produces a sine and cosine function in quadrature channels. These waveforms are applied to the x- and y-axis room-temperature (RT) shim coils by way of a modified shim controller board in a Varian XL200 spectrometer. The vector sum of these two components is a gradient rotating about the z axis of the magnet. The rotational frequency is variable from 1 to 100 Hz. The gradient intensity is adjustable by the user. It can be varied from 0 to {3.5 mT/m (60 Hz/mm 31P) in this particular implementation. An independent offset adjustment is provided. The sense of rotation of the gradient is controlled by a userselectable 1807 phase shift in the y-axis waveform. The design details of the system are described below. It is intended to be flexible for a variety of gradient experiments on highresolution and imaging NMR systems. A 16 mm phosphorus probe was used with 7 in. long, 10 mm o.d. (9 mm i.d.) outer, and 9 in. long, 5 mm o.d. inner tubes. The 16 mm saddle coil detector was located 26 mm above the bottom of the tube. The sampling region was 8 mm above and below the coil. Phosphoric acid in D2O (50%) was contained in the 10 mm tube which was placed in a sample spinner. The 5 mm tube was filled with water and held concentrically by spacers. In the Couette flow experiments, the 5 mm tube was held concentrically and stationary in the Couette flow apparatus described earlier—a Kel-F cylinder resting on screws in the upper barrel ( 9). This sample was used for all experimental spectra shown here. The response of the shim channels was mapped out at low gradient-oscillation rates (1–5 Hz) with a static sample. The two channels did not produce identical field gradients at equivalent drive voltages. This is most likely due to minor variations in component values in the existing shim controller-driver circuit. Geometric imperfections in the superconducting and RT shim coils probably play a comparable role in this phenomenon. The field components were roughly matched by measuring the width of the spectra at low rotation rates. Higher-order signal distortions were encountered due to the extra phase term encountered when the gradients are mismatched. Fine tuning was accomplished by minimizing the overtone sidebands generated with both gradient and sample rotating. A sample rotation frequency of 20 Hz and a gradient oscillation of 055 Hz were chosen to give primary sidebands at 75 Hz, secondary sidebands at 35 Hz, and tertiary sidebands from residual static gradients at 20 Hz. The sidebands at 35 Hz were monitored with variable y-axis settings at fixed x-axis settings (cf. Figs. 6 and 8). Minimum

m4277$0749

12-08-95 01:31:23

maga

FIG. 8. Spectra showing the appearance of secondary sidebands arising at spacings of the sum of the sample and gradient rotational frequencies. The sample is rotating uniformly at 20 Hz and the gradient at 055 Hz. Primary sidebands ( s ) are evident at 75 Hz spacings. Secondary sidebands ( ∗ ) are apparent at 35 Hz spacings. (Tertiary sidebands appear at 20 Hz spacings.) The intensity of the secondary sidebands decreases as the magnitudes of the orthogonal components generating the rotating gradient are brought closer. The difference is decreased from bottom (worst) to top where Gx É Gy (best match). Spectra are arranged in order of increasing Gy at fixed Gx .

intensity of the 35 Hz sidebands was taken as the best matched case. Figure 8 shows the spectra taken with a fixed x-axis setting of 250 arbitrary units. (Note that on the spectrometer used for these studies, the RT shim controls are variable from 0 to 1000 units, with a normal setting near 500 for high-resolution operation. The magnet was adjusted to achieve most of its shim corrections with the SC coils thus preserving the full range of the RT coils for experiments.) A control setting of 250 gave a gradient of 1.25 mT/m. The y axis was increased from 200 to 340 in this example. The intensity of the 35 Hz sidebands were determined in absolute value mode. Residual 55 Hz sidebands always remained after optimization which is the result of an offset between the virtual zero point of the gradients and the center of the sample. The best y-axis setting was plotted versus the x-axis setting to obtain a response function which was fitted to a linear approximation. The relation between the gradient strength and the x shim setting was determined by measuring the width of the signal at matched rotation frequencies, e.g., Fig. 4a. The change in the intensity of the sidebands arising from the mismatch in the gradient components is small. The relative magnitude of the shim misset could be measured by matching the gradient rotation frequency precisely to that of the sample and changing the sense of the gradient rotation. The mismatch component, G0 , now rotates synchronously

AP: Mag Res

158

HOPKINS, SANTINI, AND GRUTZNER

with the sample. The sidebands, at multiples of twice the spinning frequency, become spin-density projections whose width is dependent on the value of G0 . Using this model, the best effective match between the two shims was calculated to be within {0.03 mT/m. Rotating-Gradient Generator The Couette flow experiment requires that a rotating magnetic field gradient be produced. That gradient is oriented transversely to the external magnetic field. The rotational frequency of this gradient must be variable from 1 to 100 Hz to match the spinning frequency of an NMR sample tube on its air bearing. The gradient can be generated in a way that takes advantage of the normal geometry of the roomtemperature shim coils in the spectrometer. In this experiment, two sinusoidal waveforms with a 907 relative phase shift are imposed on the x- and y-axis shim coils. These particular shim coils are meant to produce a linear DC field gradient in the xy plane. An external circuit was designed such that an AC sinusoidal waveform was created in each coil shim with the required phase shift. The vector sum of these two AC gradient components is the desired rotating magnetic gradient. The circuit concept is based on a trigonometric-functiongenerator module. The trigonometric technique is necessary to produce variable-frequency, low-distortion sine waves in an exact quadrature relationship. Harmonically related frequency components of the rotating vector are suppressed by more than a factor of 1000. The actual experiment was implemented on a Varian XL-200 NMR spectrometer, but the technique is intended to be applicable to any high-resolution or imaging magnetic-resonance system. Quadrature oscillator. The circuit consists of three major sections. These are the quadrature oscillator, the trigonometric generator, and the driver circuits to the shim coils. The oscillator is fabricated with the AD-630 modulator-demodulator integrated circuit (Analog Devices, Norwood, Massachusetts). The AD-630 is a general-purpose function module, and it is capable of many modes of operation. For this experiment, the internal circuits of this module were connected as a feedback oscillator to produce a square wave output. The amplitude of the square wave output is limited to {1.8 volts by the internal design of the integrated circuit. These voltage limits set the maximum angular range of the trigonometric waveforms as described below. The frequency of the square wave is determined by an external operational amplifier that is configured as an integrator. The integrator is placed inside a circuit loop that includes the AD-630 module to produce a feedback oscillator. The oscillator circuit provides both a square wave and triangle wave output. The phase relationship between the two waveforms is fixed in this type of oscillator. If the duty cycle of the two waveforms is set to 50% the output phase of the

m4277$0749

12-08-95 01:31:23

maga

triangle wave will always be offset by {907 from that of the square wave. Trigonometric generator. The trigonometric-function generator is built with an AD-639 function generator (Analog Devices, Norwood, Massachusetts). The sine and cosine features are used, although this device is capable of generating all the trigonometric waveforms. The integrated circuit can generate an ideal function with an error level of 040 dB without additional compensation. The circuit, in this experiment, includes additional provisions to suppress these harmonics by 70 dB. The AD-639 computes an output voltage that is the ratio of a pair of independent sine waves: WÅU

F

sin(x1 0 x2 ) sin(y1 0 y2 )

G

.

[18]

The input voltages are scaled to a transfer ratio of 507 per volt. A zero to /1.8 V linear ramp produces the first quadrant of a sine wave. A constant D x or Dy input voltage will produce a fixed numerator or denominator in the above equation. The value of U scales the peak-to-peak output voltage. The experimental circuit uses two of these function-generator modules. The sine function is produced directly from the triangle waveform of the feedback oscillator. Additional circuitry was added to reduce harmonic components in the resulting sine wave to below 70 dB of the fundamental waveform. The cosine function is generated indirectly. The triangle waveform is routed to the x1 input of a second trigonometric circuit. The square wave is routed to the x2 input of this circuit. This overall configuration results in an output function that is generated by the difference between the triangle and square wave from the oscillator. This technique preserves the exact phase relationship of the original oscillator waveforms, which are offset by 907. The harmonic content of the cosine function was suppressed as described above. There is a provision that permits the user to make a 1807 phase inversion in the square waveform. This feature produces a corresponding 1807 phase shift in the cosine function. When the sine and cosine functions are added vectorially by the shim coils, the sense of rotation of the magnetic field gradient is reversed by this phase shift. Therefore, the rotating vector can be made to move in either the same or opposite sense as the turning sample tube. Shim-driver circuits. The sine and cosine signals are routed from the trigonometric circuits to the shim coils via identical circuit paths. Each of these circuit paths includes a differential analog driver–receiver circuit. The differential technique is essential to maintain the integrity of the magnet. The room-temperature shim power supply circuit common must be isolated from all other system grounds. The differential receiver prevents AC ground loop currents, particularly those that are

AP: Mag Res

MAPPING OF COUETTE FLOW USING OSCILLATING GRADIENTS

related to the power-line frequency, from circulating through the shim coils. If such loop currents are present, they will cause 60 Hz modulation artifacts to appear in spectral data. In an extreme case, the mechanical torque on the shim coils, due to the interaction of the unwanted loop current with the static magnetic field, will tear the shim coil assembly apart. Control switches are provided in several places in the circuit. One switch disables the trigonometric functions for test and adjustment of the spectrometer with the gradient generator attached. A second switch disables the DC trim voltage for the same reason. The differential circuit is disabled by a third switch. This last-mentioned control function makes the gradient-generator circuit immaterial to other users without requiring a disengagement of the gradient generator from the spectrometer. The feature is essential on any spectrometer that will be used interchangeably for conventional high-resolution applications and flow experiments. The receiver portions of the differential circuits (those parts that physically connect to the existing shim coil drivers) are constructed on a daughter-board. That subassembly is mounted on the room-temperature shim driver circuit board. These amplifiers—one set for each shim circuit—are powered from the existing isolated shim-coil analog power supply to avoid destructive ground loop currents. The rest of the vector generator is powered from a separate modular dual-output power supply. The latter power supply is isolated by the transmitter portion of the differential analog driver circuit. A detailed schematic diagram and construction details of this circuit are available from the authors. Adjustment and operation. The odd-order harmonic content of the sine and cosine functions are minimized by 5% amplitude trimming of the triangle waveform. The evenorder harmonic content is minimized by 5% trimming of the DC offset voltage of the triangle waveform. These adjustments are implemented independently for each trigonometric waveform. The optimization effect is viewed with an audio spectrum analyzer. The sine and cosine functions are monitored at the output of each trigonometric device. The harmonics in each are observed to the 10th order with the oscillator operating at 100 Hz. It is possible to adjust all unwanted harmonic content to more than 70 dB below the fundamental component of the amplitude by this technique. The adjustments are carried out iteratively for each waveform, with the overall harmonic suppression converging to a minimum value. The frequency of the oscillator (and thus the rotational rate of the gradient) is monitored with a digital frequency counter. The reference oscillator in the counter is controlled by a quartz crystal. Thus, the counter is accurate and stable to within 2 ppm. The variable frequency feedback oscillator was observed, with this counter, to be stable to within 0.1% of its operating frequency with respect to both residual shortterm frequency modulation and long-term drift.

m4277$0749

12-08-95 01:31:23

maga

159

FIG. 9. Spin-density projections. Spectrum (a) was taken with a nonspinning sample and a nearly static gradient ( õ1 Hz) of strength 3.48 mT/ m. Spectrum (b) was taken at the same gradient strength but with both sample and gradient rotating at 20 Hz.

RESULTS

Experiments were performed using two different spinning conditions: one with both inner and outer tubes rotating at the same frequency, uniform rotation without shear, and one with the inner tube held stationary while the outer tube rotated, Couette flow shearing. Uniform rotation was used to calibrate the gradient and to verify that sample rotation and gradient rotation were equivalent. Couette flow experiments were used to determine the slice selectivity and demonstrate the 2D data analysis. Uniform Rotation A series of experiments were run as a function of magnetic field and sample rotation frequencies with different magnetic field gradient strengths. The sample appears stationary in the gradient if the frequencies are matched. Figure 9 shows two such spectra. Figure 9a was taken with a static sample and the slowest gradient rotation frequency accessible, õ1 Hz. Figure 9b was taken with both the sample and gradient rotating at 20 Hz in the same sense. Both gradients were 3.48 mT/m. There is a notable difference between the widths which is attributed to line broadening from unaveraged inhomogeneity contributions in the static sample. When the magnetic field and sample rotation frequencies differ, spinning sideband patterns are generated (Figs. 7–10). Three examples are shown in Fig. 10. Figure 10a is the simulated spectrum calculated for a uniformly spinning annular sample ( Vs Å 20, Vg Å 0 Hz) in a linear transverse magnetic field gradient of 1.25 mT/m. A typical sideband pattern is ob-

AP: Mag Res

160

HOPKINS, SANTINI, AND GRUTZNER

bands that arise from mismatched gradients. Figure 7 shows the result for a sample rotating at 20 Hz with a gradient of 3.48 mT/m rotating at 083 Hz. The primary sideband set shows a 103 Hz modulation frequency. Secondary sidebands appear at 20 Hz spacing from the primary sidebands from static field inhomogeneity. The sidebands at 60 Hz spacings are combinations of second- and third-order secondary sidebands and tertiary sidebands because of gradient strength mismatch. Careful examination of the peaks at 40 and 60 Hz shows that they are split. This corresponds to the 3 Hz difference between the sample rotation frequency multiple and the sum of the two frequencies. Two-Dimensional Images of Uniform Rotation

FIG. 10. Sideband spectra showing comparison of effective rotation frequencies in a magnetic field with Gx Å Gy Å 1.25 mT/m. Spectrum (a) is the calculated spectrum for Vs Å 20 Hz, Vg Å 0 Hz ( V0 Å 20 Hz). A line-broadening function of 5 Hz was applied. Spectrum (b) is Vs Å 20 Hz, Vg Å 37 Hz ( V0 Å 17 Hz). Small artifacts are apparent due to mismatch of the x and y fields. Spectrum (c) is Vs Å 0 Hz, Vg Å 20 Hz ( V0 Å 20 Hz). The increased linewidth is due to static inhomogeneities. Note that the sideband separations in (b) are only 17 Hz compared to 20 Hz for (a, c).

served with peaks occurring at integer multiples of the spinning frequency with comparable linewidths. The simulation for Vs Å 0, Vg Å 20 Hz is identical. Spectrum 10b matches the simulation closely. It was obtained with Vg Å 37 Hz and Vs Å 20 Hz so that the peaks appear at a frequency given by the difference between the real angular frequencies. Spectrum 10c was taken with a static sample and a rotating magnetic field, Vs Å 0, Vg Å 20 Hz. While the overall pattern is similar, the lines are significantly broader and vary substantially with sideband order. The linewidths reflect the static magnetic field inhomogeneities across the sample. A rotating gradient will not average out spatial inhomogeneities across the sample. Similar linewidth effects were seen in Fig. 9. High apparent rotation rates are possible at moderate real rotation rates. Since it is possible to rotate the gradient at rates much higher than physically accessible for solution NMR experiments and the magnitudes are additive for counter rotation, it is simple to achieve apparent spinning rates of ú100 Hz. High apparent rotation rates allow better signal-to-noise ratios in large gradients since the signal can be concentrated in fewer sidebands. The sidebands are also well separated so the line positions can be accurately measured without interference from each other or the extra side-

m4277$0749

12-08-95 01:31:23

maga

A series of rotational echo spectra were taken as a function of the pulse delay t (Fig. 3b) from a sample rotating uniformly ( Vs Å 20 Hz) in a static magnetic field gradient ( Vg Å 0). This arrangement demonstrates the effects of a spinecho pulse on an ‘‘off-cycle’’ layer of fluid (e.g., Fig. 5a). Sidebands appear at 20 Hz intervals. A signal intensity versus t plot was constructed at each sideband frequency (Fig. 11a) and then Fourier transformed (Fig. 11b). The intensity

FIG. 11. Sideband intensity oscillations (a) and frequency slices (b) in the F1 dimension at the sideband (order n) frequencies (hertz) in F2 . The (real) intensity of the sideband of indicated order is measured as a function of the spin-echo delay time for a uniformly spinning sample in a 1.25 mT/m gradient. Delays corresponding to 0–1.25 rotations are shown; t delays up to 7 rotations were collected before transformation. The effective rotation frequency is 20 Hz. (b) The time evolution of each sideband is Fourier transformed to show the frequencies present in F1 . Considerable intensity is retained at zero frequency.

AP: Mag Res

MAPPING OF COUETTE FLOW USING OSCILLATING GRADIENTS

161

sideband into the second dimension. The second (12b) was taken with a 1.25 mT/m gradient rotating at 20 Hz and the sample rotating at 20 Hz. The spectral trace for F1 Å 0 now matches the static spin-density projection (Fig. 9). This confirms that the F1 Å 0 trace from the 2D analysis can be used for examining more complex flows with stroboscopic sampling. Couette Flow

FIG. 12. Two-dimensional spectra of a uniformly spinning sample (20 Hz) in (a) a static magnetic field gradient (0.62 mT/m.) and (b) a rotating magnetic field gradient (1.25 mT/m, 20 Hz). The contour plot in (a) is skewed due to the asymmetry of the imaginary component in the interferogram after the first transform. A trace at F1 Å 0 and F2 Å 0 are shown. The F1 Å 0 trace in (b) shows the static spin-density projection of the sample onto the gradient.

and phase results are shown in Fig. 11 as a function of sideband order, n (n Å 0, 1, 2, . . . , frequency offset of sideband of order n equals n*V ). Sidebands show oscillatory behavior with respect to the spin-echo delay time ( t ) which leads to a spread of frequencies along F1 . The intensity from the off-cycle layers is moved away from F1 Å 0. The major signal at F1 Å 0 comes from the layer of fluid that appears static in the rotating field. Phase cycling minimizes contributions from magnetization that has relaxed during the t delay and is placed in the transverse plane by imperfections in the 1807 pulse. The F1 Å 0 trace (top) in Fig. 12a corresponds to the Fourier transform of the signal intensity of the centerband. Figure 12 shows a pair of two-dimensional transforms as a function of echo delay for a uniform flow sample, with and without a frequency matched gradient. The first (12a) was taken with a static gradient of 0.62 mT/m and the sample rotating at 20 Hz. Individual spectral traces are shown in Fig. 12a for F1 Å 0 and for F2 Å 0. When the magnetic field is not rotating, the sidebands exhibit intensity at every position along the F1 trace, distributing the intensity of that

m4277$0749

12-08-95 01:31:23

maga

The rotating magnetic field was developed to provide radial slice selection of fluid undergoing Couette flow shearing. The selected layer of fluid appears stationary while all the other layers rotate relative to the gradient. This is demonstrated by reversing the direction of the rotating magnetic field at a frequency matched to the outer tube (Fig. 13). When the magnetic field is rotated in the same sense as the outer tube, the pattern resembles that generated by rotation of the inner tube in a static magnetic field (13). Flipping the y gradient 1807 reverses the sense of rotation of the magnetic field. The outer tube appears to have a relative frequency of twice its actual frequency, and the inner tube has a relative frequency of the magnetic field rotation frequency (half that of the outer tube’s relative frequency). This generates a pattern resembling the case where the outer tube is spinning through a static field gradient, and the inner tube is rotating at half the frequency of the outer tube. A radial frequency map was created of a fluid undergoing Couette flow shearing. A spin-echo pulse is applied which

FIG. 13. Total sideband spectra of Couette flow with the outer tube rotating at 20 Hz. A 1.25 mT/m magnetic field gradient was rotated at (a) 20 and (b) 020 Hz, demonstrating the effects of reversing the rotational direction of the magnetic field gradient.

AP: Mag Res

162

HOPKINS, SANTINI, AND GRUTZNER

FIG. 14. Traces taken at F1 Å 0 from the two-dimensional analysis of Couette flow. The outer tube was rotated at 20 Hz, the inner tube was held stationary. The gradient was set at 1.25 mT/m and its rotation frequency, Vg , varied between experiments. The separation between the intensity maxima at highest- and lowest-frequency (peaks marked l, experimental and s, calculated) scales as the diameter of the annular layer of fluid moving with the indicated frequency. Traces in column (a) are experimental; traces in column (b) are taken from full 2D simulations corresponding to the experiments.

is timed to an integral number of rotations of a particular layer of fluid which completely refocuses the magnetization from that layer (Fig. 3c). However, sidebands arising from off-cycle layers complicate the spectrum (9). In the current experiment, the two-dimensional approach was used to help alleviate the interference from these off-cycle layers. In addition to incrementing the t delay in the pulse sequence, the magnetic field rotation frequency, Vg , was adjusted in 2 or 3 Hz steps from the outer cylinder rotation frequency (20 Hz) to a minimum value of 1 Hz close to the inner cylinder. The spectral traces at F1 Å 0 for a series of Vg values are shown in Fig. 14 together with simulated spectra for the same conditions. The separation of the outer peaks in the F1 Å 0 trace gives, to a first approximation, a measure of the radial position of the fluid moving at this frequency. With a properly calibrated gradient field and fine control of the gradient rotation rate, a map of the entire sample can be made with repeated application of this process. Because of residual inhomogeneity effects, the outer peak separation was used rather than a more sophisticated matching of observed and simulated spectra. In all but the traces from the innermost layers of fluid, a sharp cutoff is observed at the outside edge of the slice. It is likely that there are harmonic contributions to the outer regions from fluid layers which

m4277$0749

12-08-95 01:31:23

maga

complete an integer number of rotations during the t delay. Contributions will appear for 10 Hz spectra and below. The effects will grow as the selected rotation frequency diminishes. The residual intensity inside the selected layer is attributed to the higher concentration of sidebands at lower frequencies in this region. Figure 15 shows a radial frequency map, generated by measuring the separation of the most intense outside peaks in Fig. 14, along with the frequency profile calculated for ideal Couette flow. Values obtained from simulated spectra are shown for comparison. The deviations at low frequency could be caused by edge effects or an eccentricity in the positioning of the inner tube. This would lead to an apparently larger inner tube (14). The observed maximum deviation is 0.15 mm corresponding to a potential 0.08 mm eccentricity. The error near the highest frequency is most likely the uncertainty in measuring the maximum near the outer wall. Signals from layers of fluid that are near in frequency contribute some intensity to the measured peaks. When these contributions are unequal on both sides of the selected cylindrical layer, the peak maximum is pulled toward the side with the most volume. Figure 16 shows expansions of the high-frequency edges of simulated F1 Å 0 traces from Couette flow experiments close to the outer wall. At magnetic field gradient rotation frequencies matching layers near the outer tube, the intensity of the outer maxima is very small. The peak intensity increases as the gradient is moved to slower frequencies. This suggests that there are substantial intensity contributions from layers rotating at frequencies close to the magnetic field gradient frequency. The signal from fluid at the outer tube is clearly smaller than that from fluid farther inside the sample. This demonstrates the addi-

FIG. 15. Plot of rotational frequency versus radius for the standard test sample undergoing Couette flow generated by the outer tube rotating at 20 Hz. The solid line is an ideal Couette flow profile. The solid dots are from the experimental traces shown in Fig. 14a. The separation between intensity maxima at highest and lowest frequency (not always the global maxima) were measured to give a separation in hertz. The radius is obtained by scaling with the magnetic field gradient strength. The open dots are the corresponding values measured from the simulated traces of Fig. 14b.

AP: Mag Res

MAPPING OF COUETTE FLOW USING OSCILLATING GRADIENTS

163

spinning rates, and reversal of spectral patterns for inner and outer tube rotation in Couette flow. A single spin-echo sequence was shown to be insufficient for layer imaging. The two-dimensional results obtained with varying echo delays show promise in significantly reducing the contribution of off-cycle layers to the spectrum of the desired layer. The velocity map demonstrates that radial position can be determined, but the slice selection does not provide a spin-density projection of the slice. More efficient cancellation of the offcycle signals is being sought. The velocity mapping accomplished here is being extended to Couette flow of non-Newtonian fluids that are not amenable to optical detection techniques. ACKNOWLEDGMENTS

FIG. 16. Expanded view of high-frequency region of Fig. 14b. Fluid near the outside tube contributes intensity to the larger peaks from fluid nearer the center. The fluid moving at the same rate as the outside tube does not give a signal strong enough to allow measurement of the gradient strength at the radius corresponding to that frequency. However, fluid moving at 12 Hz slower frequency starts to contribute a measurable peak. The maxima of the outside edge give a measure of the gradient strength for fluid moving at the indicated frequencies.

The authors acknowledge the expert assistance of the following Chemistry Departmental staff: Dr. P. J. Pellechia for the installation and testing of the electronic modifications to the NMR spectrometer; Dr. D. V. Carlson for the adaptation of the pulse sequence codes used in this experiment; Mr. M. Carlsen of the Amy Instrument Facility for circuit construction; and Mr. W. Vaughn for construction of the Couette flow apparatus. J.H. is a Department of Education National Needs Fellow. Dr. D. V. Carlson and a referee provided helpful advice for improving the manuscript. Note added in proof. We recently became aware of a paper by Shenberg and Macovski (15) which employed sinusoidal gradients in quadrature to generate a rotating gradient for fast data acquisition in MRI. They also showed that Hankel transforms could be used to process the data.

REFERENCES

tion of intensity from off-cycle layers even though the outer layers have larger physical volumes. The measured peak-topeak separation is obtained from the signal from the desired layers superimposed on signals from the ‘‘rejected’’ layers. This leads to the observed uncertainty in the derived radial separation. CONCLUSIONS

The equivalence of the NMR sideband patterns generated by rotating the sample in a static magnetic field gradient or by rotating the magnetic field gradient about a static sample has been demonstrated. Simple sideband patterns are formed provided the magnetic field is linear and homogeneous. This requires that the orthogonal components which form the rotating field are of equal strength. The artifacts that arise from a mismatch of the magnetic field components have been characterized. These artifacts are useful for optimizing the component match. Combinations of sample and magnetic field gradient rotation frequencies can be used to achieve a variety of effects: high apparent spinning rates at moderate real spinning rates, vanishing spinning rates at finite real

m4277$0749

12-08-95 01:31:23

maga

1. D. G. Cory, J. W. M. van Os, and W. S. Veeman, J. Magn. Reson. 76, 543 (1988). 2. S. A. Altobelli, R. C. Givler, and E. Fukushima, J. Rheology 35, 721 (1991). 3. Y. Xia and P. T. Callaghan, Macromolecules 24, 477 (1991). 4. K. Kose, Phys. Rev. Lett. 72, 1467 (1994). 5. M. Buszko and G. E. Maciel, J. Magn. Reson. A 107, 151 (1994). 6. C. J. Rofe, R. K. Lambert, and P. T. Callaghan, J. Rheology 38, 875 (1994). 7. G. A. Barrall, Y. K. Lee, and G. C. Chingas, J. Magn. Reson. A 106, 132 (1994). 8. G. A. Williams and H. S. Gutowsky, Phys. Rev. 104, 278 (1956). 9. J. A. Hopkins and J. B. Grutzner, J. Magn. Reson. A 111, 17 (1994). 10. R. B. Bird, R. C. Armstrong, and O. Hassager, ‘‘Dynamics of Polymeric Liquids,’’ 2nd ed., Wiley, New York, 1987. 11. G. I. Taylor, Phil. Trans. Roy. Soc. A 289 (1923). 12. R. Ernst, G. Bodenhausen, and A. Wokaun, ‘‘Principles of Nuclear Magnetic Resonance in One and Two Dimensions,’’ Oxford Univ. Press, New York, 1990. 13. M. Vera and J. B. Grutzner, J. Am. Chem. Soc. 108, 1304 (1986). 14. H. A. Snyder, Phys. Fluids 11, 1606 (1968). 15. I. Shenberg and A. Macovski, IEEE Trans. Med. Imaging MI-5, 121 (1986).

AP: Mag Res