The centrifugal viscometer

Abstract

In this study, a viscometer, which can measure the viscosity of low-volume liquids (25 μl) within 30 s, was developed on a centrifugal platform. The centrifugal viscometer consists of a disk platform and a motor. Under disk rotation, centrifugal, Coriolis, and viscosity-induced drag forces result in deflection of liquid flow. The viscosity of the liquid sample is determined by the deflection angle of the liquid, which can be examined through image analysis or visual inspection. The viscosities of a series of Newtonian model fluids were tested by the centrifugal viscometer and the results showed good agreement with the ones tested by a conventional rotational viscometer. Since the centrifugal viscometer only requires a motor to function, the microfluidic disk can be produced in large quantities at a low cost through injection molding, and the deflection angle can be detected through visual inspection, it provides an inexpensive, easy to operate, and portable approach to measure low-volume liquid viscosity.

I. INTRODUCTION

Viscosity is a material property that describes the flow characteristics of a fluid. It has been demonstrated that viscosity plays important roles in food,¹ cosmetic,² pharmaceutical,³ and chemical⁴ industries. For viscosity measurement, shear and extensional flows are the most common approaches.⁵ For shear viscosity measurement, the shear rate can be generated through either drag forces or pressure flows. The shear viscosity measuring techniques include capillary, rotational, and falling ball approaches. In the capillary technique, viscosity is determined by the relationship between flow rate and pressure drop. In the rotational technique, viscosity is determined from the relationship between torque and the rotor rotational speed. In the falling ball technique, viscosity is determined by the amount of time it takes for a ball to travel a certain distance.⁶ Currently, many viscometers have been developed, and the sample volume required for conventional viscometers is usually larger than 1 ml.

If the sample amount is limited, such as a sample from a low-birth-weight infant, a viscosity testing approach, which utilizes a sample volume much less than 1 ml, is required. Microfluidic viscometers were developed for this situation. Due to its miniature size, the microfluidic viscometer offers advantages such as a small device footprint, low sample consumption, high aspect ratio, low cost, and the ability to measure viscosities at high shear rates.⁷ Most of the micro-viscometers involved liquid flow through a micro-channel. The driving forces actuating the sample fluid include syringe pumps,^8,9 peristaltic pumps,¹⁰ centrifugal pumps,^11,12 capillary actions,^13–15 and vacuum-assisted capillary action.¹⁶ The viscosity of the sample liquid can be determined by measuring the filling time with optical sensors⁷ or timers,^11,14 measuring liquid movement with image analysis,^{10,13,15–18} analyzing the interfacial position of the fluid occupancy in a micro-channel for a test liquid and a reference liquid,^19,20 conducting signal analysis with resonant micromechanical cantilevers,^21,22 and applying the flow rate–pressure drop relationship using micro-fabricated pressure sensors²³ or a mass balance.²⁴ Although many of the approaches mentioned above have shown their ability to measure the viscosity of low-volume liquids, most of the viscosity measuring approaches require expensive instruments and complicated data analysis. Although manual time-counting approaches could reduce the cost of equipment, it is labor-intensive and prone to human error.

In order to measure the viscosity of liquids in microliters, capillary action and centrifugation are more suitable as the liquid driving forces. External pumping units, such as syringe pumps, usually require more liquid to fill the syringe and connecting pipelines. Moreover, due to the variation in surface properties, it is hard to control the flow induced by capillary action. Centrifugal pumping is superior to capillary action since it can offer a more consistent flow behavior. Centrifugal microfluidics has demonstrated its ability to actuate low-volume liquids and many fluid handling techniques such as aliquoting, metering, mixing, and washing^25–27 have been developed to assist centrifugal microfluidics to become a sample-to-result platform.^28–30 In this work, we would like to develop a micro-viscometer, which is able to measure the viscosity of low-volume liquids within a few seconds, on a centrifugal platform. The effect of the viscous drag force on the reduction of the deflection angle of the liquid is used as the mechanism for viscosity measurement. To the authors' knowledge, the idea of using the difference in the liquid deflection angle for viscosity measurement has not yet been published. The centrifugal viscometer is operated using a low-cost motor and the disk platform can be manufactured inexpensively by injection molding. In addition, the deflection angle of the liquid can be examined at a low cost through visual inspection and the total testing time is less than 30 s. The centrifugal viscometer provides an inexpensive, easy to operate, portable, and fast approach to measure low-volume liquid viscosity.

II. EXPERIMENTAL

The model fluids with viscosities ranging from 2 to 1000 cP were made by diluting corn syrup with water. Food coloring was then added to facilitate the observation of fluid flow.

The centrifugal viscometer consists of a disk platform and a spinning-control system. The disk platform is composed of two layers. The bottom layer is a polymethylmethacrylate (PMMA) disk with microfluidic structures patterned by an automatic engraving machine (EGX-350, Roland). The top layer is a flat PMMA disk. A UV-assisted solvent bonding approach³¹ was used to join both disks together. The microfluidic disk design is shown in Fig. 1(a). The inner region contains the sample reservoir and the delayed-release channel. The outer region includes a process window with many collecting chambers located at the outer radial position. The radius of the sample reservoir and the inner and outer radial positions of the process window are 3.5, 10, and 40 mm, respectively. The width and height of the collecting chamber are 2.4 and 10 mm, respectively. The spinning-control system includes a motor (HC-KFS13, Mitsubishi), an encoder (MR-J2S-10A, Mitsubishi), a motion connection accessory (MCA-7790M, National Instruments), and a motor control card (PCI-7390, National Instruments). A computer program (LabVIEW, National Instruments) was written to control the rotation of the motor. The images under high-speed rotation were captured by a complementary metal–oxide–semiconductor camera (Allied Bonito CL-400C), which is able to capture 386 frames per second at a 4 MP resolution. The photo of the experimental setup is shown in Fig. 1(c).

FIG. 1.

(a) The schematic of the disk platform for the centrifugal viscometer. (b) The force analysis on the liquid element. The flow direction is governed by the combined forces among centrifugal, Coriolis, and viscous friction forces. (c) The photo of the experimental setup. (d) The picture of the disk platform during the viscosity measurement.

For the viscosity testing, a liquid sample was loaded into the reservoir and the disk was spun to a specific rotational speed. The liquid flowed from the sample reservoir into the process window through the delayed-release channel and finally reached a specific collecting chamber according to its viscosity. The disk platform was placed upside down (with the flat disk at the bottom) to make sure that the liquid touched the surface of the flat disk while passing through the process window. As shown in Fig. 1(b), when the liquid flowed through the process window, its direction was governed by the combined forces among centrifugal ($\text{⇀}{f}_{\omega }$), Coriolis ($\text{⇀}{f}_C$), and viscous friction forces ($\text{⇀}{f}_{\mu }$). The deflection angle of the liquid element is controlled by the angular and radial motions of the liquid element. Since the angular motion of the liquid element is regulated by the Coriolis and the viscous friction forces and the radial motion of the liquid element is regulated by the centrifugal and the viscous friction forces, the deflection angle of the liquid element is influenced by the liquid viscosity. Therefore, the viscosity of the liquid can be estimated by the deflection angle of the liquid or the location of the collecting chamber where the liquid resides. Figure 1(d) shows the image of the disk platform during the viscosity measurement. If image analysis is used, a deflection angle of 70.063° is reported. On the other hand, if visual inspection is used, the liquid is residing in the 15th collecting chamber. Since the opening of the collecting chamber is about 4° and the sidewall between the collecting chamber is about 1°, a range of the deflection angle of 70°–74° is reported.

III. RESULTS AND DISCUSSION

A. Delayed-release fluidics

The deflection angle of the liquid, which results from the centrifugal, Coriolis, and viscous friction forces, is highly influenced by the rotational speed of the disk. Since it takes time for the disk to accelerate to the target rotational speed, a delayed-release channel is required in order to let the disk reach the target rotational speed before allowing the liquid to flow into the process window. A delayed-release channel with the dimensions of 0.5 × 0.5 × 14 mm³ is used in this fluidic design. Table I shows the comparison of the theoretical accelerating time (t_t), the actual accelerating time (t_a), and the lag time (t_lag) under various accelerations when the disk is rotated from 0 to 2000 rpm. As shown in Table I, the actual accelerating time is always longer than the theoretical accelerating time. This is due to the mechanical limitation of the motor since the motor requires a period of time to accelerate from a static state to a dynamic state. The lag time is defined as the time required for the liquid to flow from the sample reservoir to the process window. A low-viscosity model fluid (2 cP) is used in this experiment since it has the shortest lag time comparing to other model fluids. As shown in Table I, the lag time varies from time to time, this might have resulted from the variation of the original locations of the model fluid after loading. In order to achieve a consistent measurement of the deflection angle, the lag time must be longer than the actual acceleration time so that the liquid flows into the process window after the disk has reached the target rotational speed. As indicated in Table I, the design with the delayed-release channel can effectively increase the lag time. However, a higher acceleration (ω > 10 000 rpm/s) is also required in order to keep the lag time longer than the actual accelerating time. If a low-cost motor, which is unable to provide high angular acceleration, is used, a longer or narrower delayed-release channel is required in order to yield a longer lag time. Figure 2 shows the effect of the delayed-release channel on the liquid deflection angle when the disk is operated from 0 to 2000 rpm under an acceleration of 100 000 rpm/s. For model fluids with higher viscosities (50–1000 cP), the existence of the delayed-release design does not show much difference since the lag time is much longer than the actual accelerating time. On the other hand, for the model fluids with lower viscosities (2–10 cP), higher variations in the deflection angles are observed for the fluidic design without the delayed-release channel. Since the effect of the delayed-release channel is to provide the time required for the disk to reach the target rotational speed, in addition to the angular acceleration of the disk, the geometry (cross section and length) and the radial position of the delayed-release channel also play important roles in governing the delay time. In the following experiments, a delayed-release channel is included in the fluidic design, and the acceleration of the disk was set at 100 000 rpm/s in order to achieve consistent results.

TABLE I.

The comparison of the theoretical accelerating time, actual accelerating time, and the lag time of the liquid when the disk was operated from 0 to 2000 rpm under various accelerating conditions. Boldface denotes the operating accelerations in which the lag time is longer than the actual accelerating time.

Viscosity (cP)	Acceleration (rpm/s)	tt (s)	ta (s)	tlag (s)
				w/o delay	w/ delay
2	1000	2	2.07	1.56–1.66	1.77–1.8
2	5000	0.4	0.42	0.34–0.38	0.41–0.43
2	10 000	0.2	0.22	0.19–0.23	0 . 28–0 . 31
2	50 000	0.04	0.08	0.09–0.14	0 . 17–0 . 21
2	100 000	0.02	0.07	0.06–0.08	0 . 09–0 . 12

FIG. 2.

The effect of viscosity on the deflection angle of the liquid for the fluidic design with and without the delayed-release channel. The disk was operated from 0 to 2000 rpm under an acceleration of 100 000 rpm/s.

B. Factors that affect the deflection angle

The effect of viscosity on the deflection angle of the liquid under various rotational speeds is shown in Fig. 3. In general, the deflection angle decreases as the liquid viscosity increases, which is a result of the increase in the viscous drag force. In addition, for a constant liquid viscosity, higher rotational speeds are needed to create higher deflection angles. However, when the rotational speed exceeds 2500 rpm, flow instability occurs, and the deflection angle of the liquid is no longer related to its viscosity. Therefore, it is preferable to operate the disk viscometer at a low rotational speed range (ω ≤ 2000 rpm). Furthermore, the centrifugal viscometer shows better resolution in the low-viscosity range. As indicated in Fig. 3, a large change in the deflection angle is observed when the liquid viscosity is less than 100 cP. The deflection angle does not change much when the liquid viscosity exceeds 200 cP.

FIG. 3.

The effect of liquid viscosity on the deflection angle of the liquid under various rotational speeds. The height of the process window is 2 mm.

When the disk is operated at a high rotational speed, the Plateau–Rayleigh instability^32,33 occurs. The liquid flow changes from a continuous stream into droplets and the destination of the liquid in the collecting chamber is not related to its viscosity. Figure 4 shows schematic diagrams and photos of the liquid path in the process window under various rotational speeds. When the disk is operated at low rotational speeds (ω ≤ 2000 rpm), the liquid forms a continuous stream in the process window and flows into a specific collecting chamber [Fig. 4(a)]. Under these conditions, the deflection angle of the liquid is related to its viscosity and a very consistent angle measurement can be achieved. However, when the rotational speed of the disk reaches 3000 rpm, flow instability occurs and the liquid flow changes from a continuous stream into branched flow or droplets in the process window. The liquid flows into several collecting chambers instead of one chamber [Fig. 4(b)]. When the rotational speed of the disk is further increased to 5000 rpm, the liquid flow changes from a continuous stream into droplets. The droplets flow into the farthest collecting chamber [Fig. 4(c)]. In the last two cases [Figs. 4(b) and 4(c)], the deflection angle of the liquid cannot be used as an indicator for its viscosity. Therefore, it is more suitable to operate the centrifugal viscometer at a low rotational speed range (ω ≤ 2000 rpm) in order to avoid flow instability.

FIG. 4.

Schematic diagrams and photos of the liquid in the process window of a microfluidic disk when the disk was rotated at (a) 2000, (b) 3000, and (c) 5000 rpm. Flow instability occurs when the rotational speed exceeds 3000 rpm.

Another factor that affects the deflection angle of the liquid is the height of the process window. In the aforementioned experimental results, the height of the process window is set at 2 mm and liquid only touches one side of the process window. By reducing the height of the process window, it is possible to increase the viscous friction force by letting the liquid touch both the top and bottom surfaces. As shown in Fig. 5, when the height of the process window is below 0.1 mm, a reduction in the deflection angle is observed. However, since the reduction of the window height does not improve the resolution of the viscometer, the height of the process window was set as 2 mm for the rest of the experiments.

FIG. 5.

The effect of the liquid viscosity on the deflection angle of the liquid for various process window heights. The disk was operated from 0 to 2000 rpm with an acceleration of 100 000 rpm/s.

C. Viscosity measurement

A calibration chart for the deflection angle and the liquid viscosity using a centrifugal viscometer is shown in Fig. 6. It is clear that the increase in the liquid viscosity results in the reduction of the deflection angle. In addition, the reduction of the deflection angle is more prominent in the lower viscosity range (1–50 cP) and it gradually decreases as the viscosity increases. Therefore, the centrifugal viscometer is more effective in measuring liquid viscosities in the lower viscosity range. In addition, a good repeatability of the deflection angle measurement is also observed using image analysis. The average standard deviation for the deflection angle measurement is about 0.57° within the viscosity range of 1–1000 cP. Therefore, the testing results are very consistent.

FIG. 6.

The calibration curve of the centrifugal viscometer. The liquid viscosity can be calculated by the empirical equation of the fitting curve from the deflection angle and viscosity relationship.

To evaluate the performance of the centrifugal viscometer, the viscosity measurements of four drinks (orange juice, black fungus drink, yogurt drink, and rice milk) on the market were conducted using a commercially available viscometer (Brookfield viscometer DV-II+ Pro) and the centrifugal viscometer. The testing results are summarized in Table II. The viscosity measurements of the centrifugal viscometer were conducted by both image analysis and visual inspection. A more precise viscosity measurement can be achieved through image analysis. As shown in Table II, the testing results using the centrifugal viscometer are very close to the ones measured by the Brookfield viscometer and the deviation is within 7%. If the viscosity measurement is conducted by visual inspection, the measured viscosity can only be reported in a range of numbers since each collecting chamber contains liquid from a range of deflection angles (the opening of the collecting chamber is 4°). However, the resolution of the visual inspection approach can be further improved by reducing the size of the collecting chamber with precision machining or by enlarging the radius of the disk so that more collecting chambers can be accommodated at the outer radial position of the process window.

TABLE II.

The summary of the viscosity measurement of four drinks using the Brookfield viscometer and the centrifugal viscometer.

	Brookfield viscometer (cP)	Centrifugal viscometer (cP)
		Image analysis	Visual inspection
Orange juice	2.2 ± 0.1	2.3 ± 0.1	2–3
Black fungus drink	21.5 ± 1	21.3 ± 2.0	21–28
Yogurt drink	147.5 ± 2.5	145.4 ± 7.5	100–200
Rice milk	515 ± 5	551.5 ± 49.2	200–500

D. Cost analysis

The basic requirement for a centrifugal viscometer includes a motor and a disk platform. Since the weight of the disk platform is less than 50 g and the target rotational speed of the motor is 2000 rpm, a DC motor that costs less than 50 USD should be able to fulfill the requirement. The cost of the disk platform can be less than 1 USD if it is made by an injection molding process in mass production. If the visual inspection approach is used in measuring the deflection angle, the viscosity measurement can be carried out at a low cost. On the other hand, if a more precise measurement is required, the deflection angle can be measured by image analysis and a high-speed camera with 100 frames per second would be needed. With today's advancing manufacturing technology, a digital camera with 100 frames per second can be purchased for less than 100 USD. Therefore, the centrifugal viscometer is able to provide an inexpensive approach for liquid viscosity measurement.

IV. CONCLUSIONS

A centrifugal viscometer, which is able to measure the viscosity of low-volume liquids (25 μl), is successfully developed. A low-cost motor can be used to actuate the rotation of the disk and the viscosity of the liquid can be determined by either visual inspection or image analysis. The centrifugal viscometer is portable and affordable and can be used in many applications, especially for measuring the viscosities of liquids in limited quantities, such as the synovial fluid in the joint cavity. Although many micro-viscometers were reported to measure the viscosity of low-quantity liquids precisely, most of them rely on equipment for liquid actuation (micro-pumps) and sensing (a microscope or integrated on-chip sensors), which greatly increases the cost of the viscosity measurement. The centrifugal viscometer is able to actuate the low-volume liquid with a portable motor, and the viscosity measurement can be conducted within 30 s by visual inspection (for a rough estimation) or image analysis (for a more precise measurement). Therefore, the centrifugal viscometer offers a simple and effective way to measure the viscosity of low-volume liquid in a short time. In addition, it is suitable to measure the viscosity of bio-fluids since its most sensitive range is between 2 and 50 cP. Furthermore, by integrating the viscometer with other fluidic functions for sample preparation, it can become a sample-to-result test in many applications. Finally, most of the current sensing approaches rely on electric or optical signal changes through bio/chemical reactions. The centrifugal viscometer provides another alternative to measure the physical properties of the liquid.

ACKNOWLEDGEMENT

The authors thank the Ministry of Science and Technology of Taiwan (No. MOST 107-2221-E-035-042) for their financial support.

DATA AVAILABILITY

The data that support the findings of this study are available within the article.