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Yield Stress Measurements of Magnetorheological Fluids in Tubes Anh Dang,† Liling Ooi,† Janine Fales,‡ and Pieter Stroeve*,† Department of Chemical Engineering and Materials Science, University of California, Davis, Davis, California 95616, and Los Alamos National Laboratory, Los Alamos, New Mexico 87545
A pressure-driven apparatus was used to measure the yield stress of magnetorheological (MR) fluids as a function of the applied magnetic field, the carbonyl iron particle content, and the amount of surfactant used to stabilize the MR fluid. The yield stress was measured from the pressure difference necessary to initiate flow of a MR fluid in a straight tube present in a magnetic field. Yield stress measurements were made on MR fluids that contained up to 30 vol % carbonyl iron particles, up to 14 vol % surfactant oleic acid, and the remainder 100 cS silicone oil. In the absence of an applied magnetic field, the MR fluids did not have a yield stress and behaved as Newtonian fluids. In the presence of magnetic fields up to 2.2 kG, the MR fluids had yield stresses up to 1.9 kPa. An effect of the tube size on the yield stress was observed for MR fluids above 15 vol % carbonyl iron and magnetic fields above 1.0 kG. 1. Introduction Magnetorheological (MR) fluids are suspensions of micron-sized, magnetizable particles such as carbonyl iron, dispersed in a nonmagnetic carrier medium. The carrier medium may be varied according to the applications, but examples include silicone oil, mineral oil, and water. Surfactants such as oleic acid are usually added to stabilize the suspension by preventing the iron particles from settling. In the absence of a magnetic field, an ideal MR fluid exhibits Newtonian behavior. However, with the presence of a magnetic field, a phenomenon known as the MR effect is observed.1 The application of a magnetic field induces a magnetic dipole moment in each particle, and because of the induced dipole interactions between particles, a particle structure is formed in the fluid. The particles align into chains that are oriented parallel to the applied magnetic field. At high particle volume fractions, the chains crosslink and this causes the MR fluid to exhibit solidlike behavior with a yield stress.2 When the field is removed, the particles return to their random state and the fluid once again exhibits its original Newtonian behavior. A controllable rheological behavior is also observed in electrorheological (ER) fluids.3 ER fluids are suspensions of dielectric, nonmagnetic, solid particles in nonconducting fluids. The solid particles are polarized in the presence of an electric field and form chainlike particle structures between the two parallel electrodes. These chains also cause the development of a yield stress. The dramatic increase and reversibility of the yield stress and viscosity of these fluids with the application of an external magnetic or electric field have given rise to a variety of applications. Both ER and MR fluids are useful as damping fluids within devices such as dampers, shock absorbers, brakes, and clutches because of their controllable rheological response.4,5 However, MR fluids do offer several advantages over ER fluids. MR * To whom correspondence should be addressed. E-mail:
[email protected]. Telephone: (530) 752-8778. Fax: (530) 752-1031. † University of California, Davis. ‡ Los Alamos National Laboratory.
fluids can exhibit much higher yield stresses, approximately 2 orders of magnitude larger than ER fluids.6 Thus, MR fluids are capable of generating greater damping forces than ER fluids. Moreover, ER fluids require expensive high-voltage power supplies, while MR fluids only require small magnetic fields, which can be produced by simple, low-voltage magnetic coils. Another advantage of MR fluids is that they can operate over a broad temperature range. MR fluids have been reported to operate effectively from -40 to +150 °C.6 MR fluids have been used as resistance devices in exercise equipment and vehicle seat suspension systems. Because of the advantages of MR over ER fluids, research in MR fluids has gained recent interest. The role of magnetic saturation on the shear stresses in MR fluids was investigated using both finite element and analytical approximations by Ginder and Davis.7 MR fluids based on iron alloy particles were studied by Margida et al.,8 while Promislow and Gast2 investigated MR particle structures in a pulsed magnetic field. A number of United States patents regarding MR fluids have been published in recent years. Ginder et al.9 described a vibration damper that utilizes MR fluids. Weiss et al.10 described magnetorheological fluids with increased yield strength due to surface-modified particles. Contamination products on the surface of the particles were removed in order to enhance the yield stress. Other inventions report on MR fluids that are capable of generating large damping forces.11,12 Because of the potential commercial impact of MR fluids, there is a need to characterize fluid properties of MR fluids quickly and easily. In particular, the determination of yield stress is important for the proper design of vibration dampers. Despite the expanding interest and research in MR fluids, there is currently no commercially available device for measuring MR fluid properties. In many research laboratories, existing rheometers or viscometers have been modified to allow the incorporation of a magnetic field to the sample.13,14 The modifications typically involve constructing fixtures of low carbon steel and adding a wire coil or electromagnet to deliver a variable magnetic field to the sample.
10.1021/ie9908276 CCC: $19.00 © 2000 American Chemical Society Published on Web 06/06/2000
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Unfortunately, many of the modifications reported in the literature are complex and time-consuming and the rheometers are expensive. Consequently, in this work we report the construction of a simple pressure-driven viscometer that is inexpensive which can be used for the measurement of yield stress. We have used the apparatus to measure the yield stresses of a variety of MR fluids. The effects of the surfactant concentration and particle volume percent on the yield stress have been investigated. 2. Experimental Section Experimental Apparatus. The experimental concept was to apply a pressure to a MR fluid in a tube situated in a magnetic field until flow was initiated. The pressure necessary to initiate flow was used to calculate the yield stress. The pressure-driven viscometer consisted of the following parts: base, body and top of the vessel, stirrer, glass tubing, and electromagnet. All of the materials used in construction were nonmagnetic. The body of the vessel was constructed from Plexiglas tubing, with 8.9 cm inner diameter, 0.6 cm wall thickness, and 10.2 cm height. The base and the top of the vessel were constructed out of aluminum. An opening into the lower side of the body of the vessel was used to allow the MR fluid to flow from the vessel into a straight piece of tubing. A Swagelok connector was used to connect the tubing to the vessel. The dimensions of the different tubes used in the experiments are shown in Table 1. The tubes were 15.2 cm long, with 10.2 cm of the tube length placed inside the magnetic field. O-rings were utilized for all seals in connecting the tubes to the vessel. The top of the vessel had two openings, one to accommodate the stirrer and the other to allow the user to fill the vessel with the fluid as well as to pressurize the vessel by connecting it with a hose to a compressed nitrogen source (see Figure 1). The stirrer was a type G, Arrow air-powered stirrer with a 316 stainless steel, nonmagnetic shaft and impeller. The stirrer was connected to the laboratory air line using a hose.
Figure 1. Experimental setup.
Table 1. Dimensions of Tubes Used To Measure the Yield Stress of MR Fluids tube material
inner diameter (mm)
outer diameter (mm)
copper copper glass glass glass glass
7.9 14.0 3.8 6.0 11.0 12.5
9.5 15.9 6.4 9.5 15.9 19.1
The vessel was placed with the glass tubing between the pole faces of an electromagnet (see Figure 1). The electromagnet used was an Ealing electromagnet with 10.2 cm diameter pole faces. The gap between the pole faces was 1.9 cm. The electromagnet was connected to a power supply that was used to vary the magnetic field by varying the supplied voltage. The magnitude of the magnetic field in the gap was measured using a Bell gaussmeter (model 5070, Newark Electronics, Gaffney, SC) with a standard 5.1 cm probe. The gaussmeter provided a constant input current to the probe, which then produced an output signal that was proportional to the magnitude of the magnetic field passing through it. The accuracy of measuring the magnetic field with the gaussmeter was approximately 5%. MR Fluid Preparation. The carbonyl iron was obtained from Sigma (St. Louis, MO) with a reported particle size of 4.5-5.2 µm and a bulk density of 4.4 g/cm3. The silicone oil was also obtained from Sigma, while the oleic acid was purchased from Fisher Scientific. The MR fluid was prepared by manually stirring an amount of carbonyl iron into an amount of oleic acid. Silicone oil of 100 cS was then added. The fluid mixture was stirred carefully and constantly until a welldispersed solution was obtained. Yield Stress Measurements. The MR fluid was poured slowly into the vessel, using a glass funnel. After the MR fluid flowed out of the end of the tubing, a rubber stopper was placed into the end of the tubing to stop the flow. The tubing of the apparatus was eased carefully into the gap between the pole faces of the electromagnet. The stirrer was turned on by opening
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the air line. The speed of the stirrer was adjusted so that it was sufficient to mix the fluid without creating air bubbles. The vessel was pressurized by connecting it to a compressed nitrogen tank with a regulator to supply the desired pressure. Initially the MR fluid was allowed to flow, by removing the stopper in the tube, before the magnetic field and the pressure supply were turned on. When the magnetic field was turned on, the flow stopped and within a minute pressure was applied to the vessel in increments of 0.1-0.4 psi until flow in the tube was initiated and detected visually. The gauge pressure to initiate flow (Pflow) was recorded. To stop the flow, either the magnetic field was increased or the pressure was reduced. Identical experimental results are obtained if the process is done backward, i.e., decreasing the pressure and observing when the flow ceases. The pressure just before flow stoppage was then recorded. To calculate the yield stress of a fluid under a known pressure drop Pflow, a static force balance is performed on the cylindrical fluid element of radius r and length L. The yield stress is proportional to the pressure drop necessary to initiate liquid flow in the tube, which is
[τrz] ) rPflow/2L
°
°/
Figure 2. Yield stress measurements of Carbopol 940 and 7:3 ratio of Carbopol 940/941 as measured with glass tubes (data points) and the Weissenberg rheogoniometer (solid lines; with standard deviation, dashed lines).
(1)
Model Viscoelastic Fluids. Aqueous solutions of Carbopol 940 and 941 were used as model viscoelastic fluids in order to determine the accuracy of the pressuredriven viscometer. Carbopol 940 and 941 were obtained from Goodrich USA. These fluids have been used in various studies15-18 as model viscoelastic fluids because of the wide range of yield stresses that may be obtained by varying the type and concentration of the Carbopol in solution. Carbopol polymers as supplied are tightly coiled acidic molecules. When dispersed in water, the molecules partially uncoil because of their hydrophilic nature. The acidic Carbopol polymer has to be neutralized in order to achieve maximum thickening. Neutralization can be achieved by adding a common base, such as sodium hydroxide (NaOH), to the solution. Two model viscoelastic solutions were prepared. The first was 1 wt % Carbopol 940 in water, while the second was 1 wt % of a 7:3 ratio of Carbopol 940/941 in water. The Carbopol was dispersed in rapidly agitated water, and stirring was continued for 20 min to obtain a homogeneous solution. The solution was allowed to stand for 30 min after the mixing. The solution was slightly acidic, with a pH of approximately 3.5, and had a low viscosity. Under moderate agitation, the solution was neutralized with an 18 wt % aqueous solution of NaOH to achieve a pH between 6.5 and 7. Agitation was continued for another 2 min until neutralization was complete. Thickening occurred immediately upon addition of the NaOH. Shear stress-shear rate data for the model fluids were measured using a Weissenberg rheogoniometer equipped with two 5-in.-diameter parallel plates. The shear stress-shear rate data were extrapolated to zero shear to obtain the yield stress. The yield stresses of the fluids were also measured in the pressure-driven viscometer. 3. Results and Discussion Model Viscoelastic Fluids. The average yield stress for the Carbopol 940 solution was 160 ( 10 Pa as measured with the Weissenberg rheogoniometer. The
Figure 3. Shear stress-shear rate data of MR fluids without a magnetic field.
average yield stress from the pressure-driven viscometer from the 11.0, 6.0, and 3.8 mm glass tubes were 151 ( 11, 156 ( 11, and 163 ( 11 Pa, respectively, as shown Figure 2. In these experiments, a magnetic field was not used because the Carbopol is not magnetic. The results indicate that the measurements are within the standard deviation of the yield stress obtained with the Weissenberg rheogoniometer and that the tube diameter does not have an effect for these model viscoelastic fluids. The average yield stress for the 1 wt % 7:3 ratio Carbopol 940/941 fluid is 77 ( 8 Pa using the Weissenberg rheogoniometer. The average yield stress values obtained for the 3.8, 6.0, and 11.0 mm glass tubes are also within the standard deviation of the Weissenberg measurement as shown in Figure 2. The deviations are again independent of tube dimension. Yield stress determinations are difficult to perform, and 10% standard deviations are common in the literature. The yield stress standard deviation for the pressure-driven viscometer is only 7%, which compares favorably with the standard deviation obtained from the Weissenberg rheogoniometer, which is 7-10%. The comparisons show that the apparatus gives an accurate measurement of the yield stress. Rheological Behavior of MR Fluids. In the absence of a magnetic field, MR fluids exhibit Newtonian behavior as shown in Figure 3. The measurements were obtained with the Weissenberg rheogoniometer. The Newtonian viscosity is equal to the slope of the lines and increases as the volume percent of iron particles increases. The yield stresses for the three suspensions shown in Figure 3 are zero. Zero yield stresses were also obtained with the pressure-driven viscometer. Effect of the Volume Percent of Iron Particles. Figures 4 and 5 show that, as the magnitude of the
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Figure 6. Yield stress as a function of the surfactant concentration for MR fluids with 10 vol % iron (measured in a 11.0 mm glass tube). Figure 4. Yield stress as a function of magnetic field and volume percent iron for MR fluids with 3 vol % oleic acid (measured in a 11.0 mm glass tube).
Figure 5. Yield stress as a function of magnetic field and volume percent iron for MR fluids with 10 vol % oleic acid (measured in a 11.0 mm glass tube).
magnetic field increases from 0 to 1.9 kG, the yield stresses of the MR fluids in the 11.0 mm glass tube increase. Similar results are observed in the 6.0 and 3.8 mm glass tubes (not shown). The results indicate that as the iron volume percent increases so does the yield stress. In the absence of a magnetic field, iron particles have no permanent dipole moment and move by random Brownian motion. Application of a magnetic field induces a magnetic dipole moment in each particle, and as the magnitude of the magnetic field increases, so does the magnetic induction. Consequently, increasing the magnetic field increases the dipole forces, and that in turn increases the force required to establish flow, giving rise to a larger yield stress. At low volume fractions of iron particles, it is observed that the iron particles are aligned in single particle chains, and as the iron particle concentration increases, these chains grow in length and eventually aggregate to form thick columns.19 It is speculated that the force required to break thick columns of particles will be higher than the force required to break the structure of a single chain, leading to a larger yield stress. Effect of Surfactant Concentration. Because the density of the iron particles in the MR fluid is higher than the density of the liquid, the particles will tend to settle. The surfactant added to the suspending liquid adsorbs on the iron particle surfaces and helps to stabilize the suspension. The adsorption of oleic acid on the particles provides stearic repulsion between neighboring particles and reduces coagulation of the particles. Coagulated particles are bigger than single particles and would settle faster. Thus, oleic acid reduces the settlement of the particles. Figure 6 shows the effect of surfactant concentration on the yield stress measured
for 10 vol % iron, with 3, 6.5, and 10 vol % oleic acid, in the 11.0 mm glass tube. The results show that, as the surfactant concentration increases, the yield stress decreases. This same pattern is also observed in the 20 and 25 vol % iron MR fluids (not shown). The highest yield stresses are obtained for the MR fluids that contain 3 vol % oleic acid. This may appear to be the best surfactant concentration because higher yield stresses are obtained than with other surfactant concentrations. However, at this surfactant concentration, some sedimentation and aggregation of the iron particles were observed during the MR fluid preparation. For the 10 vol % iron concentration, settling and aggregation were not a significant problem. The suspensions are stable. At 20 and 25 vol % iron, 3 vol % oleic acid was insufficient and sedimentation and aggregation were also a problem. The suspensions with 6.5 or 10 vol % oleic acid showed good stability during the yield stress measurements: no sedimentation or aggregation was observed. From a simple calculation, one can show that the amount of surfactant present in all systems (including the 3 vol % system) is sufficient to form a packed monolayer on all iron particles if the diameter is assumed to be 5.0 µm. Because at 3 vol % the MR fluids show instability, it implies that a significant amount of the surfactant is in the silicone phase rather than on the particle surfaces. In the absence of particles, the silicone-surfactant liquids are transparent for all surfactant concentrations used here. The surfactant is soluble in the silicone oil. The experiments suggest that the amount of adsorption of surfactant on the particles is dependent on the concentration of surfactants and that beyond 6.5 vol % the particles’ surfaces are saturated with surfactant. In addition, the surfactant appears to have a dilution effect on the continuous fluid, leading to a reduction in the yield stress. The results suggest that a surfactant concentration of about 6.5 vol % is an optimum concentration if one desires high yield stresses. Constant Iron to Surfactant Ratio. The use of a fixed surfactant volume percentage does not present the same amount of surfactant per carbonyl iron particle as the volume percentage of the particles is increased. Because the presence of the surfactant is needed to adsorb on the surface of the particles, it is better to maintain a constant iron-to-surfactant ratio so that the amount of surfactant per unit surface area of particles is constant. Thus, in subsequent experiments with MR fluids, we have used a constant iron-to-surfactant ratio of 1:0.7 (vol %/vol %). For 10 vol % iron, this concentration of surfactant is close to 6.5 vol %, leading to good stability, but less than 10 vol % oleic acid used in Figure 6. The 10 vol % surfactant appears to cause a lessening of the yield stress, as seen in comparisons in Figures 4-6, which is not desirable.
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Figure 7. Effect of volume percent iron on the yield stress (6.0 mm glass tube).
Figure 8. Effect of the volume percent iron on the yield stress (11.0 mm glass tube).
Figure 9. Effect of glass tube sizes for a 10 vol % Fe/7 vol % oleic acid MR fluid.
Results for the yield stress for MR fluids with 15 vol % Fe/10.5 vol % oleic acid and with 20 vol % Fe/14 vol % oleic acid are shown in Figure 7 in the 6.0 mm glass tube. Figure 8 shows the yield stress data for identical MR fluids as in Figure 7, except that the yield stress measurements are given for the 11.0 mm glass tube. Comparisons of the two figures indicate that the yield stress for the MR fluid with 20 vol % iron measured in the 11.0 mm glass tube is lower than the yield stress measured for the same fluid in the 6.0 mm glass tube. The difference is only significant for a magnetic field strength larger than 1 kG. Comparisons of the yield stress data for the MR fluid with 15 vol % iron show that the results are practically identical. The effects of the glass tube size on the yield stress for MR fluids with 10 vol % Fe/7 vol % oleic acid and with 20 vol % Fe/14 vol % oleic acid are shown in Figures 9 and 10, respectively. In Figure 9 there is no tube size effect on the yield stress for the MR fluid with 10 vol % iron, up to a magnetic field of 1.2 kG. However, as the concentration of iron and the magnetic field are increased, an effect of the tube dimension on the yield stress is observed, as shown in Figure 10. However, for the larger tube sizes (11.0 and 12.5 mm), the yield stress
Figure 10. Effect of glass tube sizes for a 20 vol % Fe/14 vol % oleic acid MR fluid.
Figure 11. Effect of tube materials for a 15 vol % Fe/10.5 vol % oleic acid.
is the same. When the iron volume percent is 20 vol % or higher, the smaller tube diameters give higher yield stresses above 1 kG. This result is observed for all of the MR fluids with 20, 25, or 30 vol % (not shown). At magnetic fields below 1 kG, there is no effect of tube dimensions on the yield stress. An explanation for this effect is that, as the magnetic field increases for MR fluids with high iron concentration, the length of the particle chains increases. For smaller tubes more particle chain ends may be in contact with the tube walls than for larger diameter tubes. Further, some chains could bridge across the smaller tubes. Thus, the particle structure could be more connected to the smaller tube walls than the larger tube walls. The force required to break the structure in small diameter tubes should be higher than that in large diameter tubes, leading to a higher yield stress. From Figure 10 the true yield stress is obtained for the larger diameter tubes (11.0 and 12.5 mm), while the yield stress for the smaller tube (6.0 mm) gives the yield stress when there is an influence of geometry. Both measurements are of interest: one to obtain the true yield stress, and the other to obtain the effective yield stress in a conduit. The effect of glass or copper tubes on the yield stress of the MR fluids with a constant ratio of iron to oleic acid (1:0.7) is shown in Figure 11. The yield stresses of the 14.0 mm copper tube and 12.5 mm glass tube do not differ. For all iron volume fractions we do not observe a difference in the yield stress results. Measurements of the magnetic field (with the gaussmeter) inside and outside the glass and copper tubes were identical. There was no effect of tube size or tube material on the magnetic field strength. 4. Conclusions A simple and inexpensive pressure-driven viscometer was designed and fabricated to measure the yield stress
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of MR fluids in the presence of a magnetic field. The pressure-driven viscometer was used to measure the yield stress of a standard Carbopol fluid, and the results indicate that the measurements obtained with the apparatus are consistent with the yield stress observed in the Weissenberg rheogoniometer. The results of the experiments on MR fluids show that an increase in the volume fraction and the magnitude of the magnetic field gives rise to a larger yield stress. However, an increase in the surfactant concentration leads to a decrease in the yield stress. There is an optimum ratio of iron concentration to surfactant concentration, on the one hand, to overcome iron particle aggregation but, on the other hand, to obtain a maximum yield stress. There is no effect of the tube dimensions on the yield stress if the iron volume fraction is 15 vol % or less. Above 15 vol % iron, a tube size effect is observed for small diameter tubes if the magnetic field is above 1 kG. The results show that the pressure-driven viscometer is a convenient apparatus to measure the yield stress of MR fluids. Literature Cited (1) Rabinow, J. The Magnetic Fluid Clutch. AIEE Trans. 1948, 67, 1308-1315. (2) Promislow, J. H. E.; Gast, A. P. Magnetorheological Fluid Structure in a Pulsed Magnetic Field. Langmuir 1996, 12, 40954102. (3) Winslow, W. M. Induced Fibration of Suspensions. J. Appl. Phys. 1949, 20, 1137-1140. (4) Jolly, M. R.; Carlson, J. D.; Munoz, B. C. A Model of the Behavior of Magnetorheological Materials. Smart Mater. Struct. 1996, 5, 607-614. (5) Carlson, J. D.; Catanzarite, D. M.; St. Clair, K. A. Commercial Magnetorheological Fluid Devices. Int. J. Mod. Phys. B 1996, 10, 2857-2865. (6) Weiss, K. D.; Carlson, J. D.; Nixon, D. A. Viscoelastic Properties of Magneto- and Electro-Rheological Fluids. J. Intell. Mater. Syst. Struct. 1994, 5, 772-775.
(7) Ginder, J. M.; Davis, L. C. Shear Stresses in Magnetorheological Fluids: Role of Magnetic Saturation. Appl. Phys. Lett. 1994, 65, 3410-3412. (8) Margida, A. J.; Weiss, K. D.; Carlson, J. D. Magnetorheological Materials Based on Iron Alloy Particles. Int. J. Mod. Phys. B 1996, 10, 3335-41. (9) Ginder, J. M.; Elie, L. D.; Davis, L. C. Magnetic Fluid-Based Magnetorheological Fluids. U.S. Patent 5,549,837, 1996. (10) Weiss, K. D.; Carlson, J. D.; Nixon, D. A. Magnetorheological materials utilizing surface-modified particles. U.S. Patent 5,578,238, 1996. (11) Carlson, J. D.; Chrzan, M. J. Magnetorheological fluid dampers. U.S. Patent 5,277,281, 1994. (12) Carlson, J. D.; Chrzan, M. J.; James, F. O. Magnetorheological fluid devices. U.S. Patent 5,284,330, 1994. (13) Laun, H. M.; Willenbacher, K. C. Rheometry on Magnetorheological (MR) Fluids. Rheol. Acta 1996, 35, 417-432. (14) Ginder, J. M.; Davis, L. C.; Elie, L. D. Rheology of Magnetorheological Fluids: Models and Measurements. Int. J. Mod. Phys. B 1996, 10, 3293-3303. (15) Liddell, P. V.; Boger, D. V. Yield Stress Measurement with the Vane. J. Non-Newtonian Fluid Mech. 1996, 63, 235-261. (16) Atapattu, D. D.; Chhabra, R. P.; Uhlherr, P. H. T. Creeping Sphere Motion in Herschel-Bulkley Fluids: Flow Field and Drag. J. Non-Newtonian Fluid Mech. 1995, 59, 245-265. (17) Hariharaputhiran, M.; Subramanian, R.; Shankar, R.; Campbell, G. A.; Chhabra, R. P. The Settling of Spheres in a Viscoplastic Fluid. J. Non-Newtonian Fluid Mech. 1998, 79, 8797. (18) Wu, J.; Thompson, M. C. Non-Newtonian Shear-Thinning Flows Past a Flat Plate. J. Non-Newtonian Fluid Mech. 1996, 66, 127-144. (19) Mohebi, M.; Jamasbi, N.; Liu, J. Simulation of the Formation of Nonequilibrium Structures in Magnetorheological Fluids Subject to an External Magnetic Field. Phys. Rev. E 1996, 54, 5407-5413.
Received for review November 15, 1999 Revised manuscript received March 21, 2000 Accepted March 21, 2000 IE9908276
