WO2015121601A1

WO2015121601A1 - Rotational rheometry

Info

Publication number: WO2015121601A1
Application number: PCT/GB2014/050413
Authority: WO
Inventors: Richard Paul Scullion
Original assignee: Malvern Instruments Limited
Priority date: 2014-02-12
Filing date: 2014-02-12
Publication date: 2015-08-20

Abstract

The invention relates to methods and apparatus for measurement of viscosity of liquids using a rotational rheometer. Embodiments disclosed include a method of measuring viscosity of a liquid sample using a rotational rheometer ( 100) comprising first and second parts (101,102) having opposing respective first and second measurement surfaces (103, 104), a motor (105) connected to the second part (102) for rotating the second part ( 102) relative to the first part (101) and a rotational position sensor (106) for measurement of rotation of the second part (102) relative to the first part (101), the method comprising the sequential steps of: disposing the liquid sample (107) between the first and second measurement surfaces (103,104); applying a driving signal corresponding to a requested torque to the motor (105) to rotate the second part (102) relative to the first part (101) in a first direction; recording rotation of the second part (102) with the rotational position sensor ( 106) and a derived measure of torque while the second part (102) rotates in the first direction; reversing the driving signal to rotate the second part (102) relative to the first part (101) in a second opposing direction; recording rotation of the second part (102) with the rotational position sensor (106) and a derived measure of torque while the second part (102) rotates in the second direction; and calculating a measure of viscosity of the liquid sample (107) using a combination of the rotational position over time and the derived measure of torque in the first and second directions.

Description

ROTATIONAL RHEOMETRY

The invention relates to methods and apparatus for measurement of viscosity of liquids using a rotational rheometer.

Background to the Invention

Viscosity is a measure of the resistance of a material to flow under an applied force or stress, typically in the form of a shearing force. It is an important material property for many manufacturers and needs to be measured accurately during research, development and manufacture of materials and for quality control. In the case of a controlled stress rotational rheometer device, the material provides a resistive force against an applied torque, limiting the rotational speed of the device . By measuring the relationship between torque and speed, the viscosity η₀ of the material can be calculated: σ (ή T_A (f) , ri_o =→ = ^C t (equation 1) where T_A(t) is the applied torque at time t, ω₀(ί) is the steady state rotational speed (in radians per second) at the applied torque, σ is the shear stress (in Nm^"2 or Pa), f(t) is the shear rate (in s^"1) and C is a characteristic coefficient dependent on the measurement geometry.

A rheometer may also be capable of operating in a controlled strain mode. The relationship between torque and rotational speed for controlled strain rather than controlled torque is the same, but the applied torque is varied by a position controller to maintain a set speed.

In a typical configuration, a sample is sandwiched between a static, usually temperature controlled, plate and a rotatable element. The rotatable element may for example be in the form of a plate, a truncated cone or a cylinder, depending on the type of measurement required and the material. The rotatable element is connected to a motor to apply a torque to the material, thereby shearing it. Alternative configurations may involve a static element with a rotating plate, for example as provided in the Bohlin VOR rheometer (available from Malvern Instruments Limited, Malvern, UK) . The rate at which the rotatable element moves relative to the plate is measured, or controlled by varying the torque, and is used to compute the viscosity based on the known physical properties of the measurement geometry used. In either configuration, the torque is not necessarily measured at time of application, but is typically calibrated against drive current and other parameters to ensure that a desired torque is delivered. If the torque calibration drifts then an incorrect torque will be experienced by the sample, leading to an error in the value of viscosity measured.

Typically, the bearing supporting the rotatable element has a bias that is dependent on an angle of rotation, which causes the bearing to add or subtract a small force to that being applied by the motor. Typically this torque bias can be calibrated out using a mapping technique, normally referred to as a torque map. However, as the applied torque reduces, small errors in the calibration become more significant. At very low torques such errors may dominate a measurement, resulting in a relatively large residual torque offset and significant measurement errors. Changes in the torque map due to environmental and instrument conditions (temperature, air pressure etc) away from the calibration conditions are also particularly difficult to compensate for.

The main factors affecting the applied torque, i.e. the torque experienced by the sample, may be expressed as: T_A = T_R + T_M - T_B (equation 2) where T_A is the applied torque (in Nm), T_R the requested torque, T_M the torque added by the torque map and T_B the torque bias applied by the bearing.

According to the above relationship, when T_M = T_B, the applied torque T_A is the requested torque T_R. In the more realistic situation where this is not the case, the two values can be combined into a single torque offset value :

^Toff = ^ΤΜ ^{~ Τ}Β (equation 3)

One type of torque mapping method currently used, for example in the Kinexus range of instruments (available from Malvern Instruments Limited, Malvern, UK), works by rotating the bearing very slowly in a known working fluid (such as air, oil etc) in the measurement direction. The torque required to counter the force applied by the bearing and sample as it rotates is recorded and used to calculate the torque offset. Using a known working fluid allows the sample contribution to be subtracted. Slower measurements result in more accurate torque maps, but are correspondingly less suitable for frequent calibration. This method is also vulnerable to drift in the torque offset both during and after determining the torque map, as well as from any perturbations on loading a sample to be measured.

It is possible to produce an estimate of the torque map at a given location round the bearing by measuring the torque required to hold a given position. This is referred to as a point map. The calibration is still vulnerable to drift with time requiring it to be repeated regularly. Also, by including a position controller in the calibration, the noise of the system is increased. This may be offset to some extent by a design of bearing that is balanced such that any offset will average out over a full rotation to remove any torque offset from the measurement. This can, however, take a considerable time and may not be practical at low torques and/or high viscosities due to the associated low rotational speeds. Long measurement times may also not be appropriate to the sample being measured, for example if the sample will change over time by drying out or by otherwise changing composition.

The effect of a drift in torque map can be seen during a table-of-shears measurement. In such a measurement, the viscosity of a material is measured at varying torques (or angular velocities) . Typically the torque will start at the lowest torque to be measured, which may be limited by the capabilities of the rheometer, and increases to higher torques with successive measurements. When such a measurement is performed on a Newtonian liquid, i.e. where the viscosity does not change as a function of shear rate, the viscosity should be constant with requested torque . During real measurements, however, the viscosity will begin to drift from the correct value at very low torques due to the torque offset of the machine .

It is an object of the invention to address one or more of the above mentioned problems.

Summary of the Invention

In accordance with a first aspect of the invention there is provided a method of measuring viscosity of a liquid sample using a rotational rheometer comprising first and second parts having opposing respective first and second measurement surfaces, a motor connected to the second part for rotating the second part relative to the first part and a rotational position sensor for measurement of rotation of the second part relative to the first part, the method comprising the sequential steps of:

disposing the liquid sample between the first and second measurement surfaces;

applying a driving signal corresponding to a requested torque to the motor to rotate the second part relative to the first part in a first direction;

recording rotation of the second part with the rotational position sensor and a derived measure of torque while the second part rotates in the first direction;

reversing the driving signal to rotate the second part relative to the first part in a second opposing direction;

recording rotation of the second part with the rotational position sensor and a derived measure of torque while the second part rotates in the second direction; and calculating a measure of viscosity of the liquid sample using a combination of the recorded rotational position over time and the derived measure of torque in the first and second directions.

A rotational speed of the second part relative to the first part may be kept equal in magnitude during recording of rotation in the first and second directions by control of the driving signal to the motor, equating to an equal strain rate being applied to the liquid during rotation in the first and second directions. Alternatively, the requested torque may be kept constant during recording of rotation in the first and second directions, equating to an equal shear stress being applied to the liquid during rotation in the first and second directions.

In the case of equal strain rate, the measure of viscosity can be derived by dividing a difference between the requested torque in the first and second directions by the constant rotational speed and multiplying by a constant dependent on a geometry of the first and second parts.

In the case of equal shear stress, the measure of viscosity can be derived by dividing the requested torque by a difference between the rotational speed of the second part relative to the first part in the first and second directions and multiplying by a constant dependent on a geometry of the first and second parts. In the case of equal strain rate, the method may also comprise calculating a measure of torque offset from a difference between the requested torque in the first and second directions. In accordance with a second aspect of the invention there is provided a method of measuring a torque offset of a rotational rheometer comprising first and second parts having opposing respective first and second measurement surfaces, a motor connected to the second part for rotating the second part relative to the first part and a rotational position sensor for measurement of rotation of the second part relative to the first part, the method comprising the sequential steps of:

disposing a liquid sample between the first and second measurement surfaces; applying a driving signal corresponding to a requested torque to the motor to rotate the second part relative to the first part in a first direction at a constant rotational speed;

reversing the driving signal to rotate the second part relative to the first part in a second opposing direction at the constant rotational speed; and

calculating a measure of torque offset from a difference between the requested torque during rotation in the first and second directions. In embodiments according to either the first or second aspects, a rotational speed of the second part relative to the first part during rotation in the first and second directions is optionally less than 1 revolution per second. The rotational speed may be optionally less than 10^"1 revolutions per second, 10^"2 revolutions per second or 10^"3 revolutions per second.

In embodiments according to either the first or second aspects, rotation in the first and second directions may be carried out over a time period of 1 second or more, 10 seconds or more, and optionally over time periods of 60 seconds or more, or 300 seconds or more.

The measure of torque offset may be greater in magnitude than 10% of the requested torque in either of the first and second directions, i.e. in applications where the requested torque is very low and may even be of a comparable magnitude to the torque offset. The invention is particularly advantageous to these applications, where existing methods tend to have low accuracy and require frequent calibration. In embodiments according to either of the first or second aspects of the invention a magnitude of the requested torque may be less than 1 μΝιη, and optionally less than 1 nNm. As mentioned above, the invention is particularly advantageous for low torques, where errors resulting from any torque offset can otherwise be significant.

The torque offset according to embodiments of the invention may be defined as being a difference between a torque applied by the motor and the torque applied to the liquid sample. The torque offset is typically dependent on the bearing used for mounting the second part of the rotational rheometer.

In accordance with a third aspect of the invention there is provided a rotational rheometer comprising first and second parts having opposing respective first and second measurement surfaces, a motor connected to the second part for rotating the second part relative to the first part and a rotational position sensor for measurement of rotation of the second part relative to the first part, the rotational rheometer comprising a computerised controller configured to perform the method according to the first or second aspect of the invention. In accordance with a fourth aspect of the invention there is provided a computer program comprising instructions for controlling a rotational rheometer according to the third aspect to perform the method according to either of the first or second aspects. The computer program may be embodied on a tangible computer-readable medium such as a ROM disc, a magnetic memory such as a hard drive, or another type of non-volatile memory.

While simple viscosity measurements are usually performed in a single direction, rotational rheometers are typically also capable of measuring viscosity in either clockwise or anti-clockwise directions. Any torque offset, being directional, will reduce the applied torque in one direction while increasing it in the other. The invention uses this property to improve the accuracy of measurements taken at low torques. The lowest torque at which the correct viscosity can be measured provides an important specification for a rheometer. The methods according to the invention should be applicable to all Newtonian and non-Newtonian fluids that do not significantly shear thin or thicken at the range of torques being applied by the calibration.

The invention also provides a novel method of refining a torque offset measure that is more accurate than currently used methods. The method can be used to calibrate a rheometer prior to making more traditional measurements.

The methods of measuring viscosity and determining a torque offset can be used together to continuously update a torque map while taking measurements. This effectively counteracts the effect of any torque offset drift, thereby removing the requirement to continually re-calibrate a rheometer while taking measurements, and increasing the amount of time available for taking useful measurements.

Since a torque map can be determined with a sample loaded, any sample effect on the torque map is automatically taken into account using the method. This would not be the case in other known torque maps, such as a torque map in air. This further increases the utility of the device by removing the need to keep re-loading the sample after calibration. This reduces the amount of sample required and variations due to sample loading. When samples are expensive or difficult to produce in quantity, this can be a major benefit to the customer in addition to the time saving.

The invention may also be used in an oscillation mode, i.e. in which the rheometer is rotated in both directions repeatedly. The invention can be used in such applications to continuously refine the torque map without the requirement to capture any additional data during the experiment.

Many rotational rheometers typically already have the physical ability to take measurements in both directions and could therefore be modified to incorporate the technique according to the invention, which would provide a significant upgrade to the instrument in terms of quality of low torque measurement, possibly enabled using only a software change. Existing rheometers in the field could also be upgraded. On a rheometer which is limited in terms of low torque by the torque offset, using the method according to the invention will tend to produce more accurate, faster or more repeatable results than other known methods such as the bearing averaging method or the point mapping method. Detailed Description

The invention is described in further detail below by way of non-limiting examples and with reference to the accompanying drawings, in which: Figure 1 is a schematic drawing of an exemplary rotational rheometer; and

Figure 2 is a plot of shear viscosity as a function of applied torque obtained using an exemplary method according to the invention. Figure 1 is a schematic representation of a rotational rheometer 100 suitable for carrying out methods according to the invention. The rheometer 100 is connected to and controlled by a computer 1 10. The rheometer comprises first and second parts 101 , 102 with respective first and second measurement surfaces 103, 104 between which a liquid sample 107 is placed. In the illustrated embodiment, the first and second parts 101 , 102 are in the form of plates having planar measurement surfaces. Other types of geometries may be used, such as a cone and plate arrangement or a cylindrical arrangement, as is well known in the field of rheological measurements.

The second part 102 is rotatably mounted on a bearing, such as an air bearing, and is connected to a motor 105, which applies a torque to the second part 102 to cause it to rotate relative to the first part 101 , thereby applying a shear to the liquid sample 107. The torque applied may be determined as a function of a driving signal applied to the motor, given a pre-existing calibration of torque as a function of the driving signal. The torque may alternatively or additionally be measured directly by means of a sensor such as a strain gauge placed on the second part 102 or the first part. The computer 1 10 may be configured to provide a driving signal to the motor 105 to apply a constant torque by applying a constant driving signal or may apply a constant or otherwise predetermined torque using a feedback signal provided from a direct measure of torque.

The rotational rheometer 100 further comprises a rotational position sensor 106, which may be connected to the second part 102 for measurement of rotation of the second part 102 relative to the first part 101. Alternative configurations are also possible, for example where the first part 101 is rotatably mounted and is connected to a motor.

During measurements, the motor 105 rotates the second part 102 relative to the first part 101 , for example in the direction 108, while the rotational position sensor 106 measures the position of the second part 102 over time . The viscosity of the liquid sample 107 can then be determined from a measure of the applied torque and the rotational speed. This may be a constant nominal value, for example if a constant torque is applied resulting in a constant speed, or may be in the form of a relationship of viscosity as a function of shear rate, as determined from the rotational speed and the geometry of the first and second parts 101 , 102. A Newtonian liquid will result in a constant value of viscosity for all speeds, whereas non-Newtonian liquids will result in viscosity measurements that vary over the range of applied shear rates, and may also vary over time.

The rheometer will typically be loaded with a factory-calibrated torque map, which provides a relationship between driving signal and the expected resulting torque. This can for example be obtained by a single calibration using one of the methods already described, such as a measurement in air or using a known Newtonian liquid. Due to drift, however, such a torque map is likely to have significant errors which can lead to poor measurements, particularly at low torques and low speeds.

It is possible to run a rheometer without a pre-existing torque map by generating a flat torque map of offset zero. This is likely to be particularly effective if operating in a relatively flat region of the torque map which can be pre-calculated and stored, or when using materials with a viscosity which, when combined with the low applied torque, allow only slow velocities and hence small movements of the bearing.

The torque map is quickly refined using a fast but inaccurate method, such as a point map, to ensure that the initial stress applied is not significantly more than the intended level. This step is necessary unless the residual torque offset can be guaranteed to be significantly smaller than the applied torque . Alternatively, the experiment could be performed in strain controlled mode, which will guarantee that the desired strain is not exceeded. In a typical measurement, the rheometer 100 is loaded with a liquid material 107 whose viscosity is to be measured at low torque . The torque is first applied in a clockwise and then an anti-clockwise direction (or vice versa) and the rotational velocities recorded. This would usually give two measurements of the viscosity, both of which may contain large errors . The actual viscosity η₀ may be calculated as : σ C, T „T_A

η₀ =— =— = C— (equation 4) where η₀ is the actual viscosity of the material, T_A is the applied torque, ω₀ is the steady state rotational velocity (in radians/second) at the applied torque, Cj is the stress constant, C₂ is the strain constant, σ is the shear stress (in Pa) and γ is the strain rate (in s^"1) . The constant C is a combination of the stress and strain constants .

If we consider measuring the viscosity in first and second opposing rotational directions, combining equations 2 and 4 results in the following equations : η₀ = C — (equation 5) co₊ r)₀ = on 6)

where η_α is the actual viscosity of the material, C = Ci / C_2j T_R is the requested (demand) torque (in Nm), ω₊ is the steady state rotational velocity (rads/second) for the first direction measurement and co_ is the steady state rotational velocity (rads/second) for the second direction measurement.

Equations 5 and 6 above can be combined to provide an estimate for the torque offset:

¹ Toff = ¹TR (equation 7)

The result of the above relationship is that a torque map can be continuously updated during measurements to account for any drift, which can allow for correct viscosity measurements to be taken for as long as is required.

Applying equation 7 to equation 5 gives the result that: η₀ = 2C τ— (equation 8) ω₊ - co

This relationship allows a measure of viscosity to be determined from measurements or torque and rotational speed taken over first and second opposing rotational directions, and importantly without any existing torque map needing to be correct. Further measurements can also be taken for as long as is required without any requirement to recalibrate the instrument while the measurements are taking place. This is particularly useful for applications where viscosity measurements need to be taken at very low torques and/or very low rotational speeds. The minimum torque at which accurate measurements can be taken can be significantly improved since any drift in the torque offset, which becomes more significant for long measurements at low torques, is accounted for as part of the measurement.

Figure 2 illustrates a plot of shear viscosity as a function of torque as measured using a Malvern Instruments Kinexus rotational rheometer loaded with a liquid in the form of an oil having a standard viscosity of 1 Pas. The rectangular boxes indicate results obtained using the method outlined above, whereas the triangles indicate results obtained using measurements taken in only one direction. The torque map was refined between measurements as outlined above . As can be seen, by using the method according to the invention significant improvements in accuracy can be obtained, particularly at lower torques, in that the measured viscosities demonstrate a significantly smaller deviation from the actual viscosity. The illustrated results were obtained by consecutive repetition of the method at very small torques (of less than 10 nNm) and within a reasonable test time to minimize the possibility of measuring irrelevant sample behaviour such as drying or settling.

According to another embodiment of the invention it may be desirable to measure a sample using a fixed strain rate, rather than a fixed shear stress. The invention is equally applicable in either mode. In this alternative embodiment, the rheometer is configured and loaded with a sample as before. Instead of applying a constant positive and negative shear stress to the sample, positive and negative strain rates of equal magnitude are applied. If the rotational velocities are identical, and the viscosity is the same in each direction, but the requested torque is different, equations 5 and 6 change to the following:

^ _{= c} (^TR₊ + ^Toff ) (equat_ion 9)

^ _{= c} (^TR- + ^Toff ) (equation 10) where T_R+ and T_R. are the requested torques in the first and second opposing directions respectively, determined by controlling for a constant rotational speed ω₀. The above equations can be combined to solve for the torque offset, T_off, which is the difference between the magnitude of the two requested torques (since T_R_ will be negative) or, alternatively put, an average of the two requested torques:

^Toff = ( ^{~ Tr Tr+} ) (equation 1 1)

If the two requested torques are equal in magnitude, the torque offset is obviously zero. In all other cases, the torque offset is equal to half of the difference in the magnitudes of the requested torques during rotation in the first and second directions.

Once a value for the torque offset is obtained, this can be used to estimate an improved value for the viscosity using equation 9, thereby minimising the error produced by the torque map offset. A value for the viscosity η₀ can then be determined as:

(equation 12)

All advantages of the invention apply equally to this method as to the constant shear stress method. Other embodiments are intentionally within the scope of the invention, which is defined by the appended claims.

Claims

1. A method of measuring viscosity of a liquid sample using a rotational rheometer comprising first and second parts having opposing respective first and second measurement surfaces, a motor connected to the second part for rotating the second part relative to the first part and a rotational position sensor for measurement of rotation of the second part relative to the first part, the method comprising the sequential steps of:

disposing the liquid sample between the first and second measurement surfaces;

2. The method of claim 1 wherein a rotational speed of the second part relative to the first part during recording of rotation in the first and second directions is kept equal in magnitude by control of the driving signal to the motor.

3. The method of claim 2 wherein the measure of viscosity is derived by dividing a difference between the requested torque in the first and second directions by the constant rotational speed and multiplying by a constant dependent on a geometry of the first and second parts.

4. The method of claim 2 or claim 3 comprising calculating a measure of torque offset from a difference between the requested torque in the first and second directions.

5. The method of claim 1 wherein the requested torque is kept constant during recording of rotation in the first and second directions.

6. The method of claim 5 wherein the measure of viscosity is derived by dividing the requested torque by a difference between the rotational speed of the second part relative to the first part in the first and second directions and multiplying by a constant dependent on a geometry of the first and second parts.

7. A method of measuring a torque offset of a rotational rheometer comprising first and second parts having opposing respective first and second measurement surfaces, a motor connected to the second part for rotating the second part relative to the first part and a rotational position sensor for measurement of rotation of the second part relative to the first part, the method comprising the sequential steps of: disposing a liquid sample between the first and second measurement surfaces; applying a driving signal corresponding to a requested torque to the motor to rotate the second part relative to the first part in a first direction at a constant rotational speed;

calculating a measure of torque offset from a difference between the requested torque during rotation in the first and second directions.

8. The method of any one of claims 1 to 7 wherein a rotational speed of the second part relative to the first part during rotation in the first and second directions is less than 1 revolution per second.

9. The method of any one of claims 1 to 8 wherein rotation in the first and second directions is carried out over a time period of 1 second or more .

10. The method of claim 4 or claim 7 wherein the measure of torque offset is greater in magnitude than 10% of the requested torque in either of the first and second directions.

1 1. The method of any one of claims 1 to 10 wherein a magnitude of the requested torque is less than 1 μΝιη, and optionally less than 1 nNm.

12. A rotational rheometer comprising first and second parts having opposing respective first and second measurement surfaces, a motor connected to the second part for rotating the second part relative to the first part and a rotational position sensor for measurement of rotation of the second part relative to the first part, the rotational rheometer comprising a computerised controller configured to perform the method of any one of claims 1 to 1 1.

13. A computer program comprising instructions for controlling a rotational rheometer according to claim 12 to perform the method according to any one of claims 1 to 1 1.