How to measure viscosity

True to their name, gravimetric capillaries – as well as flow cups – rely on gravitational force as the drive. The resulting quantity is kinematic viscosity. Flow or efflux cups and gravimetric capillaries should only be used for measuring the viscosity of ideally viscous liquids.^[1]

Take a capillary with precisely specified dimensions (inner diameter, length) and an equally precise distance given by two marks. Let a known quantity of liquid flow through this capillary and measure the time the liquid level takes to travel from one mark to the other. The measured time is an indicator for viscosity (due to the velocity of flow depending on this quantity). To obtain kinematic viscosity (v = ny), multiply the measured flow time (t_f) by the so-called capillary constant (K_C). This constant needs to be determined for each capillary by calibrating the capillary, i.e. by measuring a reference liquid of known viscosity.

$$v = K _{C} {\cdot} t _{f}$$

Equation 1: Flow time of sample multiplied by the gravimetric capillary constant gives kinematic viscosity.

The flow time must not fall below a specified minimum, otherwise the flow inside the capillary will no longer be laminar.

If a flow cup is used, the principle works as described above, but instead of exact capillary dimensions, the volume of the cup and its outlet capillary need to be accurately defined. As a rule, the equations for getting viscosity from the flow time are empirically determined for each cup by way of calibration tests with viscosity reference standards.

Generally, gravimetric capillaries are made of glass. They are divided into direct-flow or reverse-flow models.
Direct-flow capillaries feature the sample reservoir below the distance marks, while in reverse-flow types the reservoir is placed above. This construction enables users to measure even opaque fluids. Some reverse-flow tubes are equipped with three measuring marks. Consequently, you get two flow times, which also gives you determinability.

With traditional glass capillary viscometers, the flow time is taken manually utilizing a stop watch, which demands full concentration and exact work from the operator. In addition, reliable thermostat baths as well as suitable bath liquids are required.

To save working time and material resources, automatic varieties of gravimetric capillary viscometers were developed. The most advanced models automatically fill, measure, clean, and dry. Furthermore, automatic performance excludes human mistakes, such as reading errors. The greater part of automated capillary viscometers features special capillaries working with less measuring volume than the original types, or covering an extended viscosity range. These capillaries are not in accordance with the original standards, but offer comparable precision.
Such automatic viscometers mainly employ liquid baths, sometimes combined with a thermoelectric system, for temperature control.

Flow cups are used for testing all kinds of coatings, but also ceramic suspensions, drilling fluids, and even hot bitumen. Depending on the specific application, a different flow cup is utilized.^[5]

Apart from capillary viscometers driven by gravity, there are also pressurized devices in use. We distinguish between three ways of applying pressure: by using a weight, a motor, or gas pressure.^[6]

$$\eta = K \cdot (\rho_b - \rho_s) \cdot t_r$$

Equation 2: Rolling time of the ball multiplied by the density difference of ball and sample and the adjustment constant K gives dynamic viscosity.

η ... dynamic viscosity [mPa, s]
K ... proportionality constant [1]
ρ_b ... ball density [g/cm³]
ρ_s ... sample density [g/cm³]
t_r ... ball rolling time [s]

$$F_G = m \cdot g = \rho \cdot V \cdot g$$ 

Equation 3: Density and dimensions of the ball influence the gravitational force acting on the ball.

m ... mass [kg]
g ... acceleration of gravity [m/s²]
ρ ... density [kg/m³]

The ball’s density plays a vital role, as it is included in the equation for the gravitational force (see Equation 3). By replacing the density of the liquid with the ball’s density in Equation 3, you obtain the buoyancy. Consequently, both density values (ρ_b, ρ_s) are required to calculate the viscosity. The proportionality constant (K) results from an adjustment with a viscosity reference standard.

Devices operating at inclination angles between 10° and 80° are usually defined as rolling-ball viscometers. If the inclination angle exceeds 80°, the instrument in question is a falling-ball viscometer. An exception to this rule is the Hoeppler viscometer, a falling-ball instrument which states an inclination angle of exactly 80°.

The Hoeppler principle (Fritz Höppler, 1897 – 1955, German chemist, who designed this falling-ball viscometer in 1933^[7]) conforms to DIN 53015 and ISO 12058. Temperature control is realized via liquid bath thermostat, while an operator manually performs measurement and cleaning.

Certain devices use other falling objects than balls, for example rods or needles. The original principle has also been varied: The so-called bubble viscometer measures the time it takes an air bubble in the fluid to rise. The oscillating piston viscometer of ASTM D7483 replaces gravity as the driving force by electromagnetic force, which draws a magnetic cylinder through the fluid.

The rolling-ball microviscometer implements a further development of the Hoeppler principle. This device replaces the falling tube by considerably smaller micro capillaries, reducing the required amount of sample volume. Depending on the sample’s viscosity, the filling quantity can be as low as 100 µL. Micro capillaries are available made of glass or of PCTFE, which even allows for testing fluids which corrode glass. Inductive sensors register the rolling time, while thermoelectric temperature control and adjustable inclination angles enhance the instrument’s flexibility.

Generally, a microviscometer is best suited for measuring liquids in the low-viscosity range.

Rotational viscometers push the upper limit of the potential measuring range further than gravity-based devices. They use a motor drive, which is significantly stronger than the earth’s gravitational force. Therefore, they are suited for measuring more highly viscous substances. The resulting quantity is dynamic viscosity, sometimes also referred to as shear viscosity.

Basically, a typical rotational viscometer is a cup containing the sample; the cup is matched with a so-called measuring bob that is placed in the substance under test. Depending on which part is driven, there are two principles:

• the Couette principle
• the Searle principle

The motor turns a measuring bob or spindle in a container filled with sample fluid. While the driving speed is preset, the torque required for turning the measuring bob against the fluid’s viscous forces is measured.

Besides the two varieties of the rotational measuring principle, there are manifold designs of measuring bobs. This diversity arises from a great number of different applications within a viscosity range from 1 mPa.s up to 4 000 000 mPa.s. Some systems are even suitable for testing inhomogeneous samples such as crunchy peanut butter, yogurt containing pieces of fruit, or building materials.

Find examples of some typical measuring bobs in the paragraph “Measuring Bobs for Rotational Viscometers”

We can further distinguish between types of rotational instruments by the way the torque is measured:

• Spring instruments
• Servo motor instrument

Spring instruments feature a two-part axis connected via a spiral spring and a pivot. The motor – usually a stepper motor – turns the upper part of the central axis. One end of the spring is fixed to it. The spring’s other end connects to the bob side of the axis. The pivot stabilizes the construction. One part of the pivot is also fixed to the motor axis, while its counterpart connects to the bob. In order to minimize friction, the counterpart only rests on one point. The rotation of the bob deflects the spring in proportion to the torque representing the sample’s viscosity. Two optical sensors detect the deflection utilizing slotted discs. In figure 8, the parts connected to the motor are marked light blue, the parts on the bob side light brown.

In case of low viscosity substances, the spring needs to be sufficiently sensitive, while for samples in the high-viscosity range a more robust spring is required. Consequently, one and the same apparatus cannot serve a wide measuring range. However, such a spring system is highly sensitive to low torque values.

This instrument type is equipped with a servo motor and a high-resolution digital encoder. The motor turns the main shaft to which the measuring bob is attached. The encoder records the rotational speed. The resulting motor torque is related to the viscous forces of the tested sample. As the motor current is proportional to the torque, the viscosity is calculated using motor current and rotational speed.

Such a device is not limited by the properties of a mechanical spring. Encoder and motor both support wider speed and torque ranges and, consequently, a wider viscosity range than a spring viscometer. The stiff axis makes the system more resistant to damage than the pivot-based system. On the other hand, the friction of the motor and its bearing reduces measurement accuracy, especially in the low-viscosity segment and at low speeds.

An extensive quantity of different measuring bobs serves to test various types of substances. Generally, the relation between bob size and sample viscosity is inversely proportional: The lower the viscosity is, the more voluminous the bob should be. However, most of these bobs come without defined geometry, which gives relative viscosity value^[10] and reduces comparability of measurement results. Nor is it then possible to calculate shear rate and shear stress.

Coaxial cylinder systems

Coaxial (or concentric) cylinder systems are absolute measuring systems, which also conform to DIN and ISO. Mainly special adapters for rotational viscometers house such systems. Usually these adapters are intended for a small quantity of sample and additionally offer some kind of temperature control. Due to precisely defined geometries of the bob and its cup, they permit the operator to compute shear rate as well as shear stress and, consequently, absolute viscosity values^[10].

$$\dot{\gamma} = {2 \cdot \omega \cdot R_C² \over (R_C² - R_b²)}$$

Equation 4: Shear rate in coaxial cylinder systems.

$$\tau = {M \over 2 \cdot \pi \cdot R_b² \cdot L}$$

Equation 5: Shear stress in coaxial cylinder systems.

The equation for the shear rate on the bob’s surface contains the dimensions of the parts and the rotational speed (also named angular velocity). Shear stress depends on the measured torque, the bob’s radius and length. According to Newton’s Law^[13], these parameters determine the dynamic viscosity.

A truncated cone means that the tip of the cone looks as if it has been cut off. Of course, the geometry still adheres to strict specifications. Gap a between cone and plate is usually in a range between 50 µm to 210 µm. The absence of a cone tip avoids abrasion between cone tip and plate and eliminates friction between cone and plate. Both could influence measuring results unfavorably: the former, by changing the system’s geometry over time, the latter by falsifying torque values. An additional advantage is that with a truncated system even materials with particles can be analyzed, provided the particles are no larger than one fifth of the gap between cone and plate.

Cone-plate and parallel-plate systems are also suited for extensive rheological (oscillatory) tests.