The Journal of
Physical Chemistry VOLUME 99, NUMBER 29, JULY 20,1995
0 Copyright 1995 by the American Chemical Society
LETTERS NMR Imaging of the Time Evolution of Electroosmotic Flow in a Capillary B. ManzJ P. Stilbs: B. Jinsson? 0. Sideman? and P. T. Callaghan**' Department of Physics, Massey University, Palmerston North, New Zealand; Department of Physical Chemistry, Royal Znstitute of Technology, Stockholm, Sweden: and Department of Physical Chemistry I , Chemical Center, Lund University, Lund, Sweden Received: April 6, 1 9 9 9
Velocity distributions in a capillary have been measured using NMR microimaging techniques, following application of pulsed electric fields in a weak ionic solution. The closed system exhibits electroosmosis in which the net fluid flow is zero. The time evolution of the electroosmotic flow profiles have been compared with the predictions of a theoretical model based on a wall slip within the Debye layer and drag of the remaining fluid. Excellent agreement is found both for the velocity distributions and for the time dependence of those distributions.
Introduction The electrophoretic migration of ions results in ionic collisions with the neutral solvent molecules which transfer momentum in the direction of ion flow. However, for an ionic solution in which the charge distribution is homogeneous, the oppositely directed flow of ions and cations causes local cancellation of such effects so that ion migration occurs against a stationary solvent. Because this cancellation is dependent on local charge neutrality, any inhomogeneity in charge distribution can drastically alter the local balance in electrophoretic drag forces. The resulting fluid transport is known as electroo~mosis.'-~ When an ionic solution is bounded by charged surfaces, the need to screen the surface results in just such a separation of ions of opposite charge. As a consequence of the concentration of counterions close to the surface and the resulting inhomogeneous distribution of mobile charges, the application of an electric field results in inhomogeneous electrophoretic currents. In particular, within the Debye screening layer where mobile counterions predominate, the fluid will experience significant drag in the direction of electrophoretic migration. Because the
* * @
Massey University. Royal Institute of Technology. Lund University. Abstract published in Advance ACS Abstracts, July 1, 1995.
0022-3654/95/2099-11297$09.00/0
Debye layer is generally so thin, this drag will result in an apparent slip of the fluid along the boundary. Outside this layer the flow profile will be determined by the hydrodynamics of a neutral fluid experiencingthe electroosmotic flow as a boundary condition. Observation of electroosmosis requires a technique in which the solvent flow can be profiled in a noninvasive fashion. Developments in nuclear magnetic resonance microscopy have made it possible to image fluid velocities down to a few microns per second, at a spatial resolution on the order of 10pm.637Thus, while it is not possible to use such a method to probe fluid motion within the Debye layer, it is possible to examine the hydrodynamic consequences of electroosmotic surface flow at larger distances, and to do so with a degree of time resolution subsequent to application of the electrophoretic pulse. In this article we report on the first such measurements of electroosmosis using such a technique, and we compare our results with theoretical predictions.
Theory Consider the case of a capillary of radius R in which electroosmotic flow develops at the wall. If the fluid is incompressible and confined, then the surface flow must be balanced by a counterflow. Under equilibrium, constant electric 0 1995 American Chemical Society
Letters
11298 J. Phys. Chem., Vol. 99, No. 29, 1995
RF Coil I
-i
i 2v
0
x
?fvk
I
4mm OD NMR Tube
-I
Bo
Platinum Electrode
(b) 0.5
t\
Electrophoresis Pulse 90Y
9%
n
90Y n
r. f.
-1
u -1.5
-1
-0.5
O
0.5
1
1.5
r /mm Figure 1. (a) Schematic electric field gradient pulse cycle which defines the frequency, Y,and duty cycle a = 2t&, used in eqs 3 and 4. (b) Theoretical velocity profiles, obtained from eq 3. Curve 1 shows the profile shortly before the completion of a 1.5 s EFP pulse (the second of the pair in Figure 2 ) . Curves 2 and 3 show the profiles at 0.25 and 0.75 s, respectively, after the completion of the second EFP pulse, in a pair of pulses each of 1.5 s duration and separated by 1.5 s. The cylindrical tube is assumed to have a diameter of 3.4 mm.
field conditions, and given a Newtonian fluid, we expect that the rate of strain, avlar, will vary linearly with radius, resulting in a quadratic velocity profile, v(r). The difference with the usual parabolic profile associate with Poiseuille flow is simply that the net flow rate is now zero and the nonslip boundary condition, v(R) = 0, is replaced by v(R) = VDS, the slip velocity associated with the electroosmotic surface drag. In consequence we may write V(I)
= -vDS(l - 2(r/R)’)
Because we shall be concemed with the development of the flow profile subsequent to the application of the electric field, we shall require a more general expression, v(r,t),where t refers to the time delay after the field pulse and the constant electric 00. Two of the present field profile is simply the limit as t authors9 have studied the phenomenon of electroosmosis in a number of possible geometries, such as outside a single plane, between two planes, and in a cylinder and the details of this theory will be presented in a separate paper. For the purpose of introducing those aspects of the theory relevant to the present experimental study, we shall sketch the important findings and expressions relating to capillary geometry.
-
Because of the uneven charge distribution in the layer of solution adjacent to the surface, the electric field exerts a force on this layer such that the layer is accelerated to an equilibrium velocity in a very short time, typically less than a microsecond. At the point where the electric potential from the surface can be neglected (certainly less than 10 Debye screening lengths, which corresponds in most cases to less than 1000 A) this
time
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‘’
phase
.)begin acquisition
n
1
I
n
R
’
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s
‘2 slice A -n Figure 2. (a) Arrangement of the electrophoresis cell in the NMR probe. (b) Spin warp pulse sequence used to phase-encode the image for velocity. Note that the EFP pulse is switched off subsequent to velocity encoding but prior to signal acquisition. Note that tl defines the time delay between the start of the EFP pulse and the start of the soft rf pulse. A further delay of 5.3 ms occurs before the start of velocity encoding.
equilibrium velocity is given
where er is the dielectric constant of the solution, EO the permittivity of free space, 17 the viscosity of the solution, E the magnitude of the electric field, and @(O) the electric potential at the surface of shear, the 5 potential. It should be noted that eq 2 depends on the Debye length being much less than the capillary radius. The layer of solution moving with this velocity then exerts frictional stress on the adjacent layer and, as time evolves and the electric field is maintained, layer after layer of the solution will be eventually set into motion. Provided no counterflow results, the infinite time velocity profile will correspond to the constant value given by eq 2 at all distances from the wall. We shall be concerned with the condition of zero net transport in which counterflow is required and the steady-state profile of eq 1 results. In the theoretical treatment we neglect times shorter than that required to establish surface layer flow and distances smaller than the “surface layer”. We also assume the steady-state conditions relevant to the experiments to be described, Le., that subsequent to the electrophoretic (EFP) pulse associated with velocity measurement, that an equal but oppositely directed pulses is applied, and that furthermore, many such pulse pairs are applied in succession as part of the imaging process. The process of counterflow is taken into account in the calculations by using a time-dependent but coordinate-independent pressure gradient along the cylinder axis. The time-dependent value of the pressure gradient is obtained from the requirement of zero
J, Phys. Chem., Vol. 99, No. 29, I995 11299
Letters
Figure 3. Velocity images obtained for a transverse section of the capillary at successively increasing values of delay time. T I . These dclays are respectively (a) SO, (b) 100, (c) 200, (d) 300, (e) 500,and (0 1000 ms. Note the asymmetric flow distribution in the image obtained at the longest time delay.
mean flow through any given cross section of the cylinder. Given these assumptions, it may be shown that the velocity at time t and at distance r from the tube center is given by
and
v(r,t) =
VDS is the velocity defined by eq 2, ber and bei are Kelvin functions, and a is the fraction of time the EFP pulses are on. (For the cycle shown in Figure la, a is given by 2tdt.) In the definition of S,, e, and 7 are the solution density and viscosity, respectively, while C C ) ~is the angular frequency of application of the pulse sequence. In the experiments to be described here, the cycle of EFP pulses differs from that shown in Figure la in that the time delay between the EFP pulse pair is not equal to the pair separation. However this delay was sufficient to for relaxation of the solution velocity to zero before the application of the next pulse pair. Figure l b shows the velocity profile at successively increasing time after the completion of the second EFP pulse in a pair of pulses each of 1.5 s duration and separated by 1.5 s. Because the delay time used in the experiments was 1.5 s, it is clear that the experimental situation corresponds to complete relaxation.
0
x[K,(ber(S,r) - (ber)) n= 1
[L,(ber(S,r) - (ber))
+ L,(bei(S,r)
+ K,(bei(S,r)
- (bei))]cos(nw,t)
+
- (bei))]sin(nco,t) (3)
where
4 L, = vDs-nn s i n k 7 x ber(S,R) - (ber)
r2
x sin -a
+
)[ber(S,R) - (ber)J2 [bei(S,,R) - (bei)I2
(“;“I
K, = vDs-nx sin -a sinp?
(4a)
x
bei(S,R) - (bei) [ber(S,R) - (her)]*
+ [bei(S,R) - (bei)]’
(4b)
S, = (Qoon/l.l)”’
Experimental Section
with
(ber) =
2 bei’(S,R) S,R
(bei) =
(5a)
2 ber’(S,R) S,R
(5b)
Electroosmosis experiments were performed at ambient temperature of 25 “C in situ in the microimaging probe of a Bruker AMX300 NMR spectrometer. A solution of 1 mM KCI in water was contained within a glass tube of 5 mm 0.d. and 3.4 mm i.d. Two platinum electrodes were sealed in the ends of the tube, the separation distance between the ends of the electrodes being 70 mm. The arrangement of the electrophoresis
Letters
11300 J. Phys. Chem., Vol. 99, No. 29, 1995 cell is shown in Figure 2a. EFP pulses were generated by applying pulses of 100 V amplitude to the electrodes using a specially constructed programmable high-voltage unit which was controlled using " I lpulses generated by the spectrometer. In each of these experiments the pulse sequence repetition time was 1.5 s and, after each data acquisition, a reversed EFP pulse of identical amplitude and duration was applied so as to inhibit gas formation in the cell. The velocity imaging experiments using a pulsed gradient spin echo (PGSE) velocity encoding gradient pulse pair applied subsequent to the application of the EFP pulse. Spin warp position encoding was used in which the EFP pulse was switched off subsequent to velocity encoding but prior to signal acquisition, as shown in Figure 2b. The slice selection pulse generated signal from a capillary cross section of 4 mm thickness while the phase-encoding and read gradient pulses imaged the plane of this cross section. The 2-dimensional image of the capillary cross section was obtained with a 10 mm field of view and 128* pixels giving a spatial resolution of 78 pm. The Fourier method used for velocity encoding has been described in detail e l ~ e w h e r e . ~By . ~successively increasing the amplitude, g, of the PGSE gradient pulses a 3-dimensional image set is acquired in which the third dimension corresponds to domain conjugate to the velocity spectrum. Fourier transfomation along this direction for each pixel of the image returns the local velocity spectrum for that pixel, and, by a simple algorithm in which the displacement of that spectrum along the velocity axis is determined, a local velocity value may be assigned. Because the velocities to be measured were on the order of tens of microns per second, it was necessary to maximize velocity sensitivity in the face of molecular Brownian motion. This is achieved by maximizing the separation time, A, between the PGSE pulse pair since rms Brownian displacements increase as A"* whereas displacements due to flow increase as A. The upper limit to A is determined by signal loss due to by spin relaxation. By using the stimulated echo sequence shown in Figure 2b the spins were subjected to TI rather than T2 relaxation over most of the separation time. To obtain the dependence of the velocity profile on delay time, t, the time z1 between the start of the EFP pulse and the start of the soft rf pulse was varied between 25 and 1000 ms in a set of separate experiments. The short time delay between the rf pulse and the first PGSE pulse was added to TI in order to provide an estimate of the time delay relevant to the velocity measurement. Because of the finite time needed to encode for velocity, it was desirable to keep this duration A smaller than z1 in order to optimize time resolution. As A was varied, it was necessary to adjust the duration and maximum amplitude of the PGSE pulses so as to retain optimal velocity sensitivity. Because the electroosmotic velocities are so small, it is important to establish that measurements are free from artifacts caused, for example, by eddy current induced phase shifts. This was done by obtaining a velocity image without the EFP pulse for each of the pulse parameter sets used in this work. The residue1 apparent velocity images, which represented only a weak perturbation close to the noise background, were nonetheless subtracted from the images obtained with EFP pulses so as to correct for any background artifacts. Results Figure 3 shows a set of velocity images obtained for a transverse section of the capillary at successively increasing values of delay time, zl. It is clear that these images demonstrate a symmetric, well-behaved velocity distribution below zl = 500 ms, but above that duration there is clear evidence of asymmetric flow in which cellular separation of
(a) 0.15 0.1 7
0.05
0)
E
B
O
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E
-0.15 -1.5
T,=15.3
ins
Tfin,sh-1053 ms
-1 -0.5
0 0.5 r /mm
1
1.5
1c
0.05
c
T,,,=105.3 111s T,,,,,=205.3 ms
0.1
-1.5
-1 -0.5
0 0.5 r Imm
1
1.5
-1 -0.5
0 0.5 r /mm
1
1.5
0.1 7
0.05
v1
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O
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-0.1
-1.5
Figure 4. Diametral profiles of velocity obtained from the images (ac) shown in Figure 3, along with the theoretical predictions of eq 3. In
each case two theoretical profiles are calculated using delay times corresponding to the start and end of velocity phase-encoding. These times are shown in the graphs for (a) tl = 50 ms, (b) tl = 100 ms, and (c) tl = 200 ms. Note that the only adjustable parameter is VDS which is obtained by fitting the data for t1 = 50 ms. This value is then used subsequently in calculating the absolute profiles shown in (b) and (c). counterflowing fluid is apparent, possibly due to convective effects. We shall therefore confine our quantitative analysis of velocity profiles to the short time regime. To compare our results with theoretical predictions, we have plotted a sequence of diametral profiles of velocity in Figure 4. In each case two theoretical profiles are shown in which the delay time f is set to the start and finish times of the velocity encoding PGSE gradient pair, respectively. Clearly we would hope that the experimental data might be represented by an appropriate mean of these curves. It should be noted that there are no adjustable parameters used in generating the theoretical profiles, save for the scaling constant, VDS. The value of this constant was determined by adjusting its value so that the theoretical profiles best agreed with the data in Figure 4a. The value so obtained, V D S = 0.095 mm s-', was then used in generating all other profiles. From the value of V D S we may calculate @(O) to be 80 mV (using the value 78.5 for cr),a value which is quite reasonable for g l a ~ s . ~ . ' ~ Conclusion
NMR microscopy has been shown to be a powerful method for monitoring the time evolution of electroosmosis. Further-
Letters more we have demonstrated that it is possible to measure these profiles with a velocity resolution of around 15 p m s-l. We have shown that electroosmotic flow velocities are of the same magnitude as the ionic electrophoretic motion but that electroosmotic velocities typically reach a steady-state distribution in capillary geometry on a time scale of 100 ms, rather than at the subnanosecond relaxation times pertinent to the ionic electrophoretic response. A remarkable aspect of the work is the good agreement between theory and experiment. That the predictions based on a hydrodynamic analysis fit so closely to both the shape and time dependence of the observed velocity profiles lends credence to NMR microscopy as a means of accurately and noninvasively monitoring electroosmotic flow.
Acknowledgment. B.M. and P.T.C. wish to acknowledge financial support from the New Zealand Foundation for Research, Science and Technology. B.M. also wishes to acknowledge financial support from the “DAAD-Doktorandenstipendium aus Mitteln des zweiten Hochschulsonderprogramms”. P.S., B.J., and 0,s. wish to acknowledge financial support from the Swedish Natural Science Research Council and the Swedish Research Council for Engineering Science. Note Added in Proof. During the preparation of this letter, we became aware of the work of D. Wu, A. Chen, and C. S.
J. Phys. Chem., Vol. 99, No. 29, 1995 11301 Johnson in which electroosmotic velocity profiles are deduced from the net velocity distribution averaged across the capillary. We are grateful to Professor Johnson for making us aware of this work, which will be submitted for publication elsewhere.”
References and Notes (1) Tikhomolova, K. P. Electro-osmosis; Ellis Horwood: New York, 1993. (2) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1989; Vols. 1 and 2. (3) Evans, D. F.; Wennerstrom, H. The colloidal domain wherephysics, chemistry and technology meet; VCH Publishers: New York, 1994. (4) Hunter, R. J. Zeta potential in colloid science: principles and applications; Academic Press: London, 1988. ( 5 ) Johnson, C. S.;He, Q.Adv. M a p . Reson. 1989, 13, 131-159. (6) Callaghan, P. T.; Eccles, C. D.; Xia, Y. J. Phys. E 1988,21, 820822. (7) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Oxford University Press: Oxford, 1991, (8) Results contained in the present article were first reported at the 35 Experimental Nuclear Magnetic Resonance Conference (ENC), Asilomar, April 10-15, 1994; book of Abstracts, page 75. (9) Jonsson, B.; Soderman, O., to be published. (10) Scales, P. J.; Grieser, F.; Healey, T. W.; White, L. R.; Chan, D. Y. C. Langmuir 1992, 8, 965. (11) Wu, D.; Chen, A.; Johnson, C. S., submitted to J. Magn. Reson.
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