2 1 Content 1 Fundamental rheological terms Introduction Definitions Shear stress Shear rate Dynamic viscosity Kinematic viscosity Factors which affect the viscosity Load-dependent flow behaviour Newtonian flow behaviour Pseudoplasticity Dilatancy Plasticity and yield point Time-dependent flow behaviour Thixotropy Rheopexy Temperature-dependent flow behaviour Flow behaviour of viscoelastic materials Viscoelastic liquids Viscoelastic solids Flow behaviour of elastic materials Strain Shear modulus Rheometry Tests at controlled shear rate (CSR mode) Viscosity-time test Viscosity-temperature test Flow and viscosity curves Tests at controlled shear stress (CSS mode) Viscosity-time test Viscosity-temperature test Flow and viscosity curves Creep and recovery test Measuring geometries in a rotation viscometer Coaxial cylinder measuring systems Dual-slit measuring systems according to DIN Cone-plate measuring systems according to ISO Plate-plate measuring systems...47
3 2 1 Fundamental rheological terms 1.1 Introduction Material scientists have investigated the flow and strain properties of materials since the 17 th century. The term rheology was first used in physics and chemistry by E.C. BINGHAM and M. REINER on 29 April 1929 when the American Society of Rheology was founded in Columbus, Ohio. Rheological parameters are mechanical properties. They include physical properties of liquids and solids which describe strain and flow behaviour (temporal variation of strain). Strain is observed in all materials and substances when exerting external forces. Rheometry describes measuring methods and devices used to determine rheological properties. If an external force is exerted on a body, its particles will be displaced relative to each other. This displacement of particles is known as strain. Type and extent of strain are characteristic properties of a body. Ideally elastic bodies undergo elastic strain if external anisotropic forces are exerted on them. The energy needed for this strain is stored and effects spontaneous full recovery of the original form if the external force ceases to act. Ideally viscous bodies undergo an irreversible strain if external anisotropic forces (e.g. gravitational force) are exerted on them. The input energy is transformed. This increasing viscous strain is known as flowing. There are only few fluids with practical importance which show (almost) ideally viscous behaviour. Most materials are neither ideally viscous nor ideally plastic. They rather exhibit different behaviour and are thus called viscoelastic materials.
4 3 The most simple model to illustrate rheological properties is the parallel plate model. The top plate, which has a surface area A [m²], is moved by a force F [N = kgm/s²] at a speed v [m/s]. The bottom plate remains at rest. The distance between the plates, to which the material adheres, is described by h [m]. Now, thinnest elements of the liquid will be displaced between the plates. This laminar flow is of fundamental importance for rheological investigations. Turbulent flows increase the flow resistance, thus showing false rheological properties. A moved plate F v h fix plate Fig. 1-1: Parallel plate model Shear rate = v / h in s -1 Shear stress τ = F / A in Pa Viscosity η = τ / in Pas Strain γ = dx / h dimensionless In addition to the expression, the symbol D is also used for the shear rate. Shear tests are usually conducted using rotation viscometers. In contrast to the parallel plate model, the moved surface performs a rotary movement.
5 4 1.2 Definitions Shear stress Force F acting on area A to effect a movement in the liquid element between the two plates. The velocity of the movement at a given force is controlled by the internal forces of the material. Shear stress τ = Force F Area A N = ( Newton) m 2 = Pa 100 Pa = 1 mbar = 1 hpa old unit: dyn / cm 2 = 0.1 Pa Shear rate By applying shear stress a laminar shear flow is generated between the two plates. The uppermost layer moves at the maximum velocity v max, while the lowermost layer remains at rest. The shear rate is defined as: Shear rate = dv dh where dv Velocity differential between adjacent velocity layers dh Thickness differential of the flow layers In a laminar flow the velocity differential between adjacent layers of like thickness is constant (dv = const., dh = const.) The differential can thus be approximated as follows: Shear rate = Velocity v Distance h = m/ s m = s -1 In addition to, the symbol D is also used for the shear rate in the literature.
6 5 Process ranges in s -1 Example of use Sedimentation of fine particles in suspensions Paints, lacquers, pharmaceutical solutions Flowing due to surface characteristics Paints, printing inks Dripping under the effect of the gravitational force Paints, coatings Extruding Polymers Chewing/ swallowing Food Spreading butter on a slice of bread Food Mixing, agitating Substances in process engineering Brushing Paints, lacquers, pastes Spraying, spreading Paints, lacquers, coatings Rubbing in Creams, lotions High-speed coating Paper coatings Lubrication of machine parts Mineral oils, greases Table 1-1: Typical shear rate ranges
7 Dynamic viscosity Viscosity describes the toughness of a material. Shear stress τ Dynamic viscosity η = Shear rate = Pa s -1 = Pas The unit Pas (or mpas) is used for the viscosity. 1 Pas = 1000 mpas old unit: 1 P (Poise) = 100 cp (Centipoise) = 100 mpas (Millipascalsecond) Substance Dyn. viscosity η in mpas Acetone 0.32 Water 1.0 Ethanol 1.2 Mercury 1.5 Grape juice Cream approx % sugar solution 57 Olive oil: approx. 100 Honey approx. 10,000 Plastic melts Tar approx Bitumen approx Earth mantle approx Table 1-2: Typical viscosities at 20 C in mpas
8 Kinematic viscosity If ideally viscous materials are tested using a capillary viscometer, such as an UBBELOHDE viscometer, the kinematic viscosity ν is determined, not the dynamic viscosity η. The kinematic viscosity is related to the dynamic viscosity through the density of the material. Dynamic viscosity η mm Kinematic viscosity ν = = Density ρ s 2 Old unit: cst (Centistokes) = mm² / s There used to be several device-specific measuring methods to determine the kinematic viscosity, e.g. FORD cups, and, accordingly, a large number of units, such as FORD cup seconds, ENGLER degrees, REDWOOD or SAYBOLD units. These viscosity-dependent values cannot be converted into absolute viscosities η or ν for non-newtonian fluids.
9 8 1.3 Factors which affect the viscosity The flow and strain behaviour of a material may be affected by a number of external factors. The five most important parameters are: Substance The viscosity of a material depends on its physical and chemical properties. Temperature Temperature has a major effect on the viscosity. For example, several mineral oils lose about 10 % of their viscosity if the temperature is only increased by 1 K. Shear rate The viscosity of most materials depends on the shear rate, i.e. the load. Time The viscosity depends on the strain history of a material, in particular on previous loads. Pressure If great pressure is exerted on a material, its viscosity may increase as particles are organised in a more tight structure (resulting in more interaction possibilities). Other influencing factors are the ph value, magnetic and electric field strength.
10 9 2 Load-dependent flow behaviour Flow and viscosity curves The flow behaviour of a material is characterised by the relation between shear stress τ and shear rate. A -τ diagram is often used for graphic representation. Usually, the shear stress is shown on the ordinate and the abscissa the shear rate, irrespective of whether or τ were given for the measurement. These diagrams are referred to as flow functions or flow curves. τ [Pa] [s -1 ] Fig. 2-1: Flow curve Plotting the viscosity over the shear rate or shear stress τ produces the viscosity function or viscosity curve. η [mpas] [s -1 ] Fig. 2-2: Viscosity curve The measuring result obtained with a viscometer or rheometer is always a flow curve. However, the viscosity function can be calculated based on the measured values.
11 Newtonian flow behaviour If a NEWTONian material is subjected to a shear stress τ, a shear gradient of viscous flow is generated which is proportional to the applied shear stress. The flow function of a NEWTONian material is a straight line which runs through the origin of the coordinate system at an angle α. This relation between shear stress and shear gradient is described by NEWTON s law of viscosity. τ η = = const. η is the material constant of the dynamic shear viscosity. If the viscosity is plotted over the shear rate (or shear stress) in a viscosity diagram, a straight line which starts at = 0 s -1 (or τ = 0 Pa) and runs parallel to the abscissa is obtained for an ideally viscous material. τ η Fig. 2-3: Flow curves of NEWTONian materials Fig. 2-4: Viscosity curves of NEWTONian materials The viscosity of a NEWTONian material or ideally viscous material is independent of the shear rate. Examples of NEWTONian materials: water, mineral oil, sugar solution, bitumen
12 11 According to NEWTON, a viscous body may be represented by the mechanical model of a damper. Force F Fig. 2-5: Mechanical model of a NEWTONian body It can be shown with the help of this model that the material is continuously deformed in the damper as long as a force acts on the piston. If the force ceases to act, the original shape is not restored. A viscous strain is characterised in that the energy input to create a flow is fully transformed into heat in an irreversible process. Materials which show only little interaction between (usually short) molecules exhibit NEWTONian flow behaviour.
13 Pseudoplasticity Many materials exhibit a strong decrease in viscosity if the shear rate grows. This effect is of great technical importance. Compared with an ideally viscous material, a pseudoplastic or structurally viscous material can be pumped through pipelines with a lower energy input at the same flow velocity. τ / = η const. The proportionality factor τ / in the NEWTONian constitutive equation is thus referred to as η a. η a is the apparent viscosity and denotes the viscosity at a certain shear rate. Mathematical expression for pseudoplastic materials according to OSTWALD DE WAELE: τ = K n where n < 1 for pseudoplastic materials Transformed into a viscosity function: η = τ = K n 1 τ η Fig. 2-6: Flow curve of a pseudoplastic material Fig. 2-7: Viscosity curve of a pseudoplastic material Examples: suspensions, dispersions, paints, lacquers, creams, lotions, gels Materials are referred to as pseudoplastic if a force acting on the body causes the particle size to change, the particles to be oriented in the direction of flow, or an agglomerate to be dissolved.
14 Dilatancy The viscosity of dilatant materials also depends on the shear rate. It increases as the shear rate grows. Dilatant behaviour can cause trouble in technological processes. Mathematical expression according to OSTWALD DE WAELE: τ = K n where n > 1 for dilatant materials Transformed into a viscosity function: η = τ = K n+ 1 τ η Fig. 2-8 Flow curve of a dilatant material Fig. 2-8 Viscosity curve of a dilatant material Dilatant behaviour is found rather seldom. Examples: concentrated corn starch dispersions, wet sand, several ceramic suspensions, several surfactant solutions Note: If in case of high shear rates the flow in the measuring gap is no longer laminar, but becomes turbulent, this may falsely suggest dilatant behaviour.
15 Plasticity and yield point Plasticity describes structurally viscous liquids which have an additional yield point τ o. A practical example of such a material is toothpaste. At rest, toothpaste establishes an network of intermolecular bonding forces. These forces prevent individual volume elements to be displaced when the material is at rest. If an external force which is smaller than the internal forces (bonding forces) acts on the material, the resulting strain is reversible, as with solids. However, if the external forces exceed the internal bonding forces of the network, the material will start flowing, the solid turns into a liquid. Definition of the yield point τ o Maximum shear stress τ at the shear rate = 0 s -1 Thus, if F external < F internal, the material does not flow if F external > F internal, the material starts to flow Examples: Toothpaste, PVC paste, emulsion paint, lipstick, fats, printing ink, butter Flow curves and mathematical expression of materials with a yield point Flow curves of plastic liquids do not start in the origin of the coordinate system, but run on the ordinate axis until the yield point τ o is reached, then they converge from the ordinate. The flow curves can be expressed mathematically using a number of equations, depending on the actual material. For example, the flow curve of chocolate is typically based on the CASSON model. Mathematical expression according to BINGHAM: τ = f B + η B (where f B = yield point according to BINGHAM) Mathematical expression according to CASSON: τ = f C + η C (where f C = yield point according to CASSON) Mathematical expression according to HERSCHEL and BULKLEY: τ = f H + m p (where f H = yield point according to HERSCHEL and BULKLEY) where p < 1 for pseudoplastic and p > 1 for dilatant materials
16 15 τ τ τ o τ o Fig. 2-10: Flow curve according to BINGHAM Fig. 2-11: Flow curve according to HERSCHEL and BULKLEY Physical causes for the occurrence of yield points in dispersions are intermolecular particle-particle and particle-dispersing agent interactions. - VAN DER WAALS forces - Dipole-dipole interactions - Hydrogen bonds - Electrostatic interactions The ST. VENANT model (with static friction) is a mechanical model used to describe plastic behaviour. Fig. 2-12: ST. VENANT model for plastic materials
17 16 3 Time-dependent flow behaviour 3.1 Thixotropy Thixotropy is a property exhibited by non-newtonian liquids, they return to their original viscosity only with a delay after the shear force ceased to act. In addition, these materials often also have a yield point. Tomato ketchup is an example of such a material. When stirred or shaken, ketchup becomes thinner and only returns to its original viscosity after allowing to rest for a while. Per definition, a thixotropic material does not only thin depending on the shear rate, but it additionally returns to its original viscosity after a material-specific period of rest. Theses gel-sol and sol-gel changes in thixotropic materials are reproducible. Yoghurt serves as a counter-example: it becomes thinner when stirred, but does not return to its original thickness. Yoghurt thus does not exhibit thixotropic flow behaviour. η Shear at = const. Sample at rest ( = 0.1 s -1 ) Fig. 3-1: Viscosity-time curve of a thixotropic material t Two transitional areas can be easily identified in the viscosity-time curve shown above. A gel is quickly transformed into a sol at a constant shear force. During the period of rest the materialspecific network structures are re-established, i.e. the sol turns back into a gel.
18 17 τ UP η UP DOWN DOWN Fig. 3-2: Flow and viscosity curve of a pseudoplastic and a thixotropic material The flow curve shows that the measured rising and declining curves are not congruent. The area between the two curves (hysteresis area) defines the extent of the time-dependent flow behaviour. The larger the area the more thixotropic is the material. Examples: paints, foodstuff, cosmetics, pastes
19 Rheopexy Rheopectic materials exhibit greater viscosity while they are subject to shear stress. Structures are established in the material during the application of mechanical shear forces. The original viscosity is only restored with a delay after the shear forces ceased to act on the material, by disintegrating this structure. η Shear at = const. Sample at rest ( = 0.1 s -1 ) Fig. 3-3: Viscosity-time curve of a rheopectic material t This process of viscosity increase and decrease can be repeated as often as you wish. In contrast to thixotropy, true rheopexy is very rare. τ η UP UP DOWN DOWN Fig. 3-4: Flow and viscosity curve of a pseudoplastic and a rheopectic material Examples: several latex dispersions, several casting slips, several surfactant solutions
20 19 4 Temperature-dependent flow behaviour As mentioned above, the viscosity of a material is a function of temperature. Exact temperature control and accurate indication of the measuring temperature is thus of major importance in viscosity measurements. The viscosity-temperature curve of a material is found at a constant shear rate. In most materials, the viscosity decreases as the temperature is raised. In ideally viscous materials, this phenomenon can be described with the help of the ARRHENIUS equation: η = A e B T where T... temperature in Kelvin A, B.. material constants η T Fig. 4-1: Viscosity-temperature curve Viscosity-temperature curves are also often established in order to trace certain reactions. For example, the curing temperature can be determined for powder lacquers (see Fig. 4-2, curve a), or the chocolate melting process can be followed (see Fig. 4-2, curve b). η a b T Fig. 4-2: Viscosity-temperature curves of certain reactions
21 20 5 Flow behaviour of viscoelastic materials Viscoelastic materials exhibit both viscous and elastic properties. Because of physical differences, a distinction is made between viscoelastic liquids and viscoelastic solids. The elastic portion of viscoelastic liquids is described by HOOKE S law, which is also known as the spring model, the viscous portion by NEWTON S damper model. 5.1 Viscoelastic liquids Viscoelastic materials and purely viscous materials can only be distinguished if they are stirred, for example. Rheological phenomena cannot be observed while they are at rest. In a viscous liquid, a rotating agitator causes centrifugal forces, which drive volume elements of the liquid towards the wall of the cup. This means that a dip is formed around the agitator shaft. In contrast, in elastic liquids the rotating agitator causes normal forces which are so great that they do not only compensate the centrifugal forces but exceed them. Consequently, volume elements of the liquid are thus dawn up the agitator shaft. This phenomenon of masses creeping up a rotating shaft due to the acting normal forces is called the WEISSENBERG effect. Fig. 5-1: Flow behaviour of a viscous and a viscoelastic liquid A force acting on a viscoelastic material (see Fig. 5-2) causes the spring to be deformed immediately, but the damper reacts with a delay. If the force ceases to act, the spring will return immediately while the damper remains displaced, so that the material partly remains strained. That means that there is no full restoration of shape. The amount of the spring return corresponds with the elastic portion, the amount of the remaining strain (damper) corresponds with the viscous portion. A viscoelastic liquid can thus be described using a model where damper and spring are arranged in series. Honouring JAMES C. MAXWELL ( ), this series connection is also called the MAXWELL model.
22 21 Fig. 5-2: Maxwell model Examples of viscoelastic liquids are gels and silicone rubber compounds 5.2 Viscoelastic solids If a force is exerted on a viscoelastic solid a delayed strain will take place, because the displacement of the spring is impeded by the damper. If the force ceases to be exerted, the body will fully return to its original shape (but again delayed by the damper). That means that there is a full restoration of shape. It can thus be said that a viscoelastic solid is characterised in that it has the ability of reversible strain. The model to describe such materials is again a combination of spring and damper models. In contrast to viscoelastic liquids, however, the two elements are connected in parallel, as in the KELVIN-VOIGT model. Fig. 5-3: KELVIN-VOIGT model An example of a viscoelastic solid is hard rubber.
23 22 6 Flow behaviour of elastic materials 6.1 Strain A cube with an edge length h shall be investigated here as the volume element in order to illustrate the strain γ. The bottom face of the volume element is fixed. A force F acts on its top face, which is thereby displaced by the amount s. F s h h α Fig. 6-1: Unloaded volume element Loaded volume element Mathematical expression: s tan α = = γ h Strain is a dimensionless parameter. A deformation angle α of 45 corresponds with a strain γ of 1 or 100%. The symbol for the shear rate ( ) can be derived from that for the strain (γ). The shear rate describes the strain change dγ during a period of time dt. Consequently, is the derivative of strain γ with respect to time t. In other words, the shear rate can be considered to be the strain rate. 6.2 Shear modulus In purely elastic bodies, the ratio of shear stress τ and resulting strain γ is constant. This materialspecific parameter described the stiffness of the material and is known as the shear modulus G. Shear stress τ Pa Shear modulus G = = = Pa Strain γ 1 The shear modulus can be in the MPa (10 6 Pa) range in very stiff bodies.
24 23 7 Rheometry Viscometers or rheometers used to determine the rheological properties are called absolute viscometers if the measured values are based on the basic physical units of force [N], length [m] and time [s]. dynamic viscosity η = [N/m 2 ] [s] = force / length 2 time = [Pa] [s] Viscometers are measuring devices which are used to determine the viscosity depending on the rotational speed (= shear rate), time and temperature. Rheometers are devices which are additionally able to determine the viscous and viscoelastic product properties depending on the force (shear stress) exerted in both rotation/ creep test and oscillation test. Absolute viscometers have the advantage that the results of the measurements are independent of the device manufacturer or measuring equipment used. The material under investigation is filled into the two-part measuring system, where a shear force is applied. The measurements must be conducted under certain boundary conditions in order to obtain correct results. The applied shear force must only produce a stratified laminar flow. Vortices and turbulences in the measuring system consume much energy so that viscosity values up to 40 % above the true viscosity are obtained. The material under investigation should be homogeneous. It is not recommended to mix substances during the measurement. Due to the different specific weights of the components a phase separation of the mix may occur during the measurement. The changed composition of the mix in the measuring gap may lead to erroneous viscosity values. The material must adhere to the walls. The force exerted on the top plate must be fully transmitted on to the sample. This condition is not fulfilled if the boundary layers of the material under investigation do not properly adhere to the parts of the measuring system. Problems in this respect may be encountered with materials such as fats. The elasticity of the sample must not be too high. The shear forces acting on the sample may otherwise result in extreme normal forces such that the material creeps out of the measuring gap. When handling elastic materials, the shear rate must be chosen such that normal stress does not affect the measuring result. The shear stress applied according to the laws of viscosity is proportional to the shear rate and shall only be as great as is necessary to maintain stationary flow conditions, i.e. a flow at constant velocity. The force that is necessary to accelerate or decelerate the flow is not considered in this equation. The NEWTONian relation η = τ / shows that the viscosity η can be determined using either of two different measuring methods.
25 Tests at controlled shear rate (CSR mode) (CSR controlled shear rate) With this measuring method, the rotational speed n is preset at the rheometer, and the shear rate is calculated based on the gap h and the rotational speed n (or circumferential speed v of the shearing area). The flow resistance moment M (or the shear force F) of the braking, tough material under investigation is measured. This torque M is converted into the rheological parameter of shear stress using the shear area A of the measuring system. physical setting rheological setting rot. speed n [min -1 ] shear rate [s -1 ] measurement physical result torque M [mnm] rheological result shear stress τ [Pa] η = τ The dynamic viscosity η is calculated from the shear stress τ and shear rate (or D).
26 Viscosity-time test A constant shear rate is applied for a certain period of time. The shear stress τ is measured as a function of time. The viscosity determined according to the viscosity equation is obtained in relation to time. This test is used in practice in stability investigations, to study hardening reactions and with thixotropic materials. [s -1 ] t [s] Fig. 7-1: Conditions: shear rate = const., temperature = const. τ [Pa] 3 η [Pas] t [s] t [s] 1 Material with a viscosity which does not change over time (e.g. calibration oil) 2 Material with a viscosity which decreases over time (e.g. ketchup) 3 Material with a viscosity which increases over time (e.g. hardening lacquer) Fig. 7-2: Results of the viscosity-time tests
27 Viscosity-temperature test At a constant shear rate, a temperature ramp is preset and the viscosity is measured in relation to temperature. In practice, this test is conducted when it comes to investigating temperaturedependent hardening or melting reactions. T [ C] Fig. 7-3: Conditions: shear rate = const., variable temperature t [s] τ [Pa] 2 η [Pas] T [ C] T [ C] 1 Material with a viscosity which decreases as the temperature rises (e.g. chocolate) 2 Material with a viscosity which increases as the temperature rises (e.g. hardening lacquer) Fig. 7-4: Results of the viscosity-temperature tests
28 Flow and viscosity curves At a constant temperature, a shear rate - time profile is preset. The shear stress is measured for each shear rate value and the corresponding viscosity is calculated from those measuring results. [s-1 ] T=const. t [s] Fig. 7-5: Conditions: variable shear rate, temperature = const. A constant viscosity value is found in materials with ideally viscous behaviour (NEWTONian materials), such as water. In materials with pseudoplastic behaviour the viscosity decreases as the shear rate rises ( shear thinning ). In contrast, in materials with dilatant behaviour the viscosity increases as the shear rate rises ( shear thickening ). τ [Pa] η [Pas] [s -1 ] [s -1 ] 1 NEWTONian material 2 Pseudoplastic material 3 Plastic material 4 Dilatant material Fig. 7-6: Flow and viscosity curves of materials with different rheological behaviour
29 28 Flow and viscosity curves are often plotted such that the viscosity values are shown at both rising and falling shear rate. Between the two sections of the curve (up ramp, down ramp) there is often a section where the shear rate is kept constant. [s -1 ] Fig. 7-7: Conditions t [s] In addition to load-dependent flow behaviour (pseudoplasticity, dilatancy), the results of the measurements then also allow information about time-dependent flow behaviour to be derived. In practice, the area between the up and down curves is often used as a measure for time-dependent flow properties, i.e. for the thixotropy of the material. Materials which require a long time after maximum shear to re-establish their structure show in the diagram a large area between the two curves (hysteresis area). τ UP η UP DOWN DOWN Fig. 7-8: Flow and viscosity curve of a pseudoplastic and a thixotropic material
30 29 In order to be able to compare measured curves with respect to the hysteresis areas, an identical measuring profile must be selected for the individual measurements, i.e. the duration of the individual test phases, the total test duration, the shear rate profile and the temperature must be identical. In addition to pseudoplastic behaviour, thixotropic behaviour can also facilitate certain materials to be processed. Thanks to the reduced viscosity over time, there is less power required for pumping, mixing, spraying or brushing. An advantage of thixotropic coatings is that they spread more smoothly after application and bubbles which are possibly created during the application process can escape more easily. Pseudoplasticity and thixotropy are two rheological properties which exist fully independent of each other. They should not be mistaken.
31 Tests at controlled shear stress (CSS mode) (CSS controlled shear stress) In controlled shear stress tests, the torque M is preset at the rheometer, and the shear stress τ is calculated from the torque M and the shear area A of the measuring system. The rotational speed n achieved by the plunger due to the applied torque is measured. This rotational speed n is converted into the rheological parameter of shear rate using the appropriate measuring system factor. physical setting torque M [mnm] rheological setting shear stress τ [Pa] measurement physical result rot. speed n [min -1 ] rheological result shear rate [s -1 ] η = τ The dynamic viscosity η is calculated from the shear stress τ and shear rate.
32 Viscosity-time test A constant shear stress τ is applied to the material under investigation for a certain period of time. The achieved shear rate is measured as a function of time. The viscosity determined according to the viscosity equation is obtained in relation to time. This test is applied in practice when studying hardening reactions. It boasts an advantage over the controlled shear rate test: during hardening the viscosity of the material increases and at a constant shear stress the shear rate falls. This gradually reduces the foreign movement which interferes with the hardening process. τ [Pa] t [s] Fig. 7-9: Conditions: shear stress = const., temperature = const. [s -1 ] 2 η [Pas] t [s] t [s] 1 Material with a viscosity which does not change over time (e.g. calibration oil) 2 Material with a viscosity which decreases over time (e.g. ketchup) 3 Material with a viscosity which increases over time (e.g. hardening lacquer) Fig. 7-10: Results of the viscosity-time tests
33 Viscosity-temperature test At a constant shear stress, a temperature ramp is preset and the viscosity is measured in relation to temperature. In practice, this test is conducted when it comes to investigating temperaturedependent hardening or melting reactions. T [ C] τ = const. t [s] Fig. 7-11: Conditions: shear stress = const., variable temperature [s -1 ] 1 η [Pas] T [ C] T [ C] 1 Material with a viscosity which decreases as the temperature rises (e.g. chocolate) 2 Material with a viscosity which increases as the temperature rises (e.g. jellying starch) Fig. 7-12: Results of the viscosity-temperature tests
34 Flow and viscosity curves At a constant temperature, a shear stress - time profile is preset. The shear rate is measured for each shear stress value and the corresponding viscosity is calculated from those measuring results. τ [Pa] T=const. t [s] Fig. 7-13: Conditions: variable shear stress, temperature = const. A constant viscosity value is found in materials with ideally viscous behaviour (NEWTONian materials), such as water. In materials with pseudoplastic behaviour the viscosity decreases as the shear stress rises. In contrast, in materials with dilatant behaviour the viscosity increases as the shear stress rises. τ [Pa] η [Pas] [s -1 ] [s -1 ] 1 NEWTONian material 2 Pseudoplastic material 3 Plastic material 4 Dilatant material Fig. 7-14: Flow and viscosity curves of materials with different rheological behaviour:
35 34 In mathematics, the given parameter is usually plotted on the abscissa, and the resulting parameter on the ordinate. In contrast, in rheology is plotted on the abscissa and τ on the ordinate, irrespective of whether shear rate or shear stress are given for the measurement. The shear stress test is the only measuring method where the yield point can be determined metrologically. Only if the applied shear stress exceeds the network bonding forces in the material under investigation the material will start to flow, i.e. a measurable shear rate is obtained. In the flow curve shown in Fig. 7-6, material 3 exhibits a yield point. The flow curve does not pass the origin of the coordinate system, but shows a certain translation on the Y axis. In plastic materials an additional shear stress is required in order to obtain a shear rate in the material. Also in controlled shear stress tests, flow and viscosity curves are often plotted such that the viscosity values are shown at both rising and falling shear stress. Between the two sections of the curve (up ramp, down ramp) there is often a section where the shear stress is kept constant. τ [Pa] T=const. t [s] Fig. 7-15: Conditions In addition to the yield point value (if any), the results of the measurements then also allow information about load-dependent flow behaviour (pseudoplasticity, dilatancy), and time-dependent flow behaviour (thixotropy, rheopexy) to be derived.
36 35 τ UP η UP DOWN DOWN Fig. 7-16: Flow and viscosity curve of a pseudoplastic and a thixotropic material In order to be able to compare measured curves with respect to the hysteresis areas, which provides information about the thixotropic or rheopectic behaviour of a material, an identical shear rate range should be selected for the individual measurements, and the duration of the individual test phases, the total test duration and the temperature must be identical. As various shear rates can result from a shear stress setting, a shear stress controlled measurement is less suitable for comparing hysteresis areas than a shear rate controlled test. The amount of thixotropy and rheopexy is determined from the flow curves by finding the difference between the areas under the up and down sections of the curve. Thixotropic materials have positive, rheopectic materials negative values.
37 Creep and recovery test The creep test forms a simple and quick method used to find the viscoelastic properties of the material under investigation (with exact distinction between the viscous and elastic portions). The mobile part of the measuring arrangement is loaded with a constant shear stress (τ x ) for a certain period of time (t 0 bis t 2 ). The sample reacts on this force with a deformation, i.e. the material starts to creep. In the second part of the test (t 2 bis t 4 ), the material is relieved from the shear stress so that it can recover. τ [Pa] τ x t 0 t 2 t 4 t [s] Fig. 7-17: Creep and recovery test Ideally elastic materials A piece of vulcanised rubber will be now be looked at as an example for an ideally elastic material. The constant shear stress applied leads to a certain twist in the sample, i.e. it shows the strain γ. The angle of such strain is characterised by the spring modulus of elasticity of the purely elastic solid. Shear stress and resulting strain γ show a linear relation. If the force is doubled, the strain will also double. HOOKE s spring model serves as a model for ideally elastic materials. The strain is maintained as long as the deforming force keeps acting. 100 % of the strain energy is stored in the spring. The sample will be 100 % relieved if the force ceases to be exerted.
38 37 γ t 0 t 2 t 4 t [s] Fig. 7-18: Creep and recovery curve of an ideally elastic material (according to HOOKE) Ideally viscous material Serving as an example for a material with ideally viscous behaviour, water shows a completely different behaviour. The constant shear stress applied leads to a strain γ which increases linearly over time, i.e. it shows flowing. The input energy is used up for the flowing process. If the sample is relieved, the strain γ obtained by this moment of time will be maintained. NEWTON s damper model is used to describe an ideally viscous material. γ t 0 t 2 t 4 t [s] Fig. 7-19: Creep and recovery curve of an ideally viscous material (according to NEWTON)
39 Viscoelastic liquids and solids The reaction of viscoelastic liquids on applied shear stress shows characteristics of both elastic and viscous strain. A partial recovery by the elastic portion γ E can be observed, but the portion of viscous strain γ V remains. γ γ E γ V t 0 t 2 t 4 t [s] γ E Elastic portion (recovery) γ V Viscous portion Fig. 7-20: Creep and recovery curve of a viscoelastic liquid In a viscoelastic solid, a delayed but complete recovery can be observed, i.e. γ V is almost zero. A material shall now be scrutinised which consists of macromolecules (parallel connection of spring 2 and damper 2) linked by springs (spring 1) in a highly viscous oil (damper 3). This model is also known as the BURGER model. Fig. 7-21: BURGER model
40 39 The applied shear stress initially leads to a spontaneous jump in strain (dilatation of spring elements 1 which are situated in the orientation of the strain), the strain rate then drops. During that time the macromolecules are oriented, the twisted springs tensioned and the macromolecules stretched up to their mechanically maximal possible size (delayed viscoelastic strain of spring 2 and damper 2). If more force is applied, the strain will again increase linearly if the macromolecules are irreversible disentangled and caused to flow with the viscous matrix mass (viscous strain of damper 3). If the test duration is long enough, all dampers and springs finally show maximum dilatation. During the relieve phase, two types of recovery take place. Spring 1 returns to its original tension immediately (elastic recovery), and the parallel connection of spring 2 and damper 2 recovers with a delay (viscoelastic recovery). Damper 3 remains fully displaced, so that a partial strain is maintained. If this remaining strain is very small (γ V near 0 %), the material is called a viscoelastic solid, otherwise it is a viscoelastic liquid. Creep curve (t 0 to t 2 ) Recovery curve (t 2 to t 4 ) γ 1... Purely elastic strain (spring 1) γ 2... Viscoelastic strain (parallel connection of spring 2 and damper 2) γ 3... Purely viscous strain (damper 3) γ max Maximum strain: γ 1 + γ 2 + γ 3 β.. Gradient angle of the strain curve on achievement of the stationary flow condition (depending on the viscosity of the strained material) γ E Elastic recovery portion Viscous recovery portion γ V Fig. 7-22: Creep and recovery curve with analysed parameters
41 40 In the load phase, unstationary flowing occurs in the linearly viscoelastic range between the points of time, t 0 and t 1. The viscosity here depends on the applied shear stress and time. ( τ,t) η = f x Stationary viscous flowing is observed between the points of time, t 1 and t 2. The viscosity does no longer depend on the elapsed loading time. ( ) η = f τ x The zero viscosity η 0 corresponds with the behaviour of damper 3 in the BURGER model (see Fig. 7-13). It can be determined as η 0 = τ γ x 3 = τ x tan β where the shear rate γ 3 = dγ / dt = tan β If determined at smallest shear rates, the zero viscosity η 0 is a material constant, which contains information e.g. about the molecular weight of non-networked macromolecules. In creep tests a constant shear stress is applied and the time-dependent strain is measured. Mathematically, the relation between stress and strain can be expressed as follows: γ( t) = J(t) τ x This equation introduces the time-dependent compliance factor J(t). Like the zero viscosity, it is a material-specific quantity. It is a measure of the softness or flexibility of a material. The greater the compliance the more can the material be strained under the application of a certain shear stress. J(t) = γ(t) / t x in [Pa]
42 41 8 Measuring geometries in a rotation viscometer The measuring systems used in a rotation viscometer usually consist of a rotating and a rigid part. The rotating plunger is turned at either a preset speed or a preset torque. Two measuring principles are distinguished in coaxial measuring systems: (1) The SEARLE principle Rotating plunger and resting cup (or bottom plate) (2) The COUETTE principle Rotating cylinder (or bottom plate) and resting plunger Measuring systems which employ the COUETTE principle have drawbacks when it comes to temperature control, because it is more difficult technically to seal a rotating face than a fixed one. This is why the use of COUETTE systems is often restricted to low speeds. Fig. 8-1: Cylinder measuring arrangement- Cylinder measuring arrangement according to the SEARLE principle according to the COUETTE principle
43 Coaxial cylinder measuring systems DIN describes the coaxial measuring system. Coaxial means that rotating member and resting member of the measuring system are disposed on one rotation axis. Such cylinder measuring systems are also known as concentric measuring systems. The terms of the parallel plate model can be applied to the round cylinder, if its surface area is idealised as many small plane faces. v(r) R i R a R i... Radius of the plunger (inner cylinder) R a... Radius of the cup (outer cylinder) r... Distance of a layer of the liquid from the rotation axis v(r)... Distribution of the circumferential speed in the measuring gap Fig. 8-2: Cross-section through a Searle cylinder measuring system The shear stress distribution τ(r) and the shear rate distribution (r) are dependent on the radii of the measuring system. In order to obtain near-linear distributions (r) and τ(r), the measuring gap must not be too large. DIN thus specifies a maximum ratio of radii δ = R a /R i In other words, the DIN standard defines the ratio of radii, but not the absolute radii or gap size. The ratio of radii δ = R a /R i is specified in the DIN standard to be 1.1, preferably
44 43 Coaxial cylinder measuring system according to DIN α L.. Length of the plunger (inner cylinder) L.. Immersed skirt length L.. Distance between bottom edge of plunger and cup base R i... Radius of the plunger (inner cylinder) R a.. Radius of the cup (outer cylinder) R s... Radius of the plunger skirt α... Opening angle of the plunger cone Fig. 8-3: Cylinder measuring system according to DIN / ISO 3219 DIN prescribes the following geometrical arrangement for cylinder measuring systems: Ratio of radii δ = R a /R i 1.1 (preferably ) 90 α 150 (preferably 120 ± 1 ) L / R i 3 (preferably 3.00) L 1 / R i 1 (preferably 1.00) L 2 / R i 1 (preferably 1.00) R s / R i 0.3 Calculation of the shear stress τ from the torque M (measuring systems according to DIN 53019) τ = M 3 R i M in [mnm], R i in [m], τ in [Pa] Calculation of the shear rate from the rotational speed n (measuring systems according to DIN 53019) = n n in [min -1 ], in [s -1 ]
45 Dual-slit measuring systems according to DIN This special coaxial cylinder measuring system with a particularly large shear area was standardised to be able to investigate materials which have a very small viscosity (such as waterbased lacquers, for example). The plunger has the shape of a tube and the cup has a cylindrical core section. This dual-slit measuring system thus takes advantage of two shear faces, an inner and an outer plunger surface. R 4 R 3 R 2 L R 1 Fig. 8-4: Dual-slit measuring system according to DIN According to DIN 54453, ratio of radii: δ = R 4 / R 3 = R 2 / R immersed length: L 3 R 3
46 Cone-plate measuring systems according to ISO 3219 In the cone-plate measuring system, the material to be investigated is disposed between the bottom plate and the measuring cone. According to DIN 53018, the cone angle must be rather small in order to allow the simplified expression tan β = β (β in rad) to be applied. The cone angle of that measuring system is chosen such that for each point on the cone surface the ratio of angular speed and distance to the plate is constant. This means that there is a constant shear rate across the entire radius of the measuring cone. The tip of the cone must just touch the bottom plate in this geometry. In order to prevent the cone tip from wear and at the same time to provide the possibility to measure materials which contain fillers, most rheometer manufacturers lift the cone tip by a certain amount ( µm). Measuring cones with an angle of 1 are most widespread. However, in order to be more flexible when it comes to measuring materials which contain fillers, dispersions are often measured using cones with an angle of 4. This guarantees a laminar flow to be generated in the measuring gap despite the dispersed particles. As a rule of thumb, a laminar flow can be assumed as long as the particle diameter is five times smaller than the gap. In addition to a constant shear rate across the entire measuring gap, the cone-plate measuring system has further advantages, such as high shear rates, small sample quantities and easy cleaning. In order to obtain accurate results of the measurements, it is of major importance to conduct them very carefully. The measuring cone must be adjusted such that its imaginary tip just touches the bottom measuring plate. Another point is that of filling the measuring system. It is filled correctly if the material under investigation is visible approx. 1 mm around the entire circumference of the cone. The material must not escape from the measuring gap or rest on top of the measuring cone during the measurement. flattened cone tip correct filling approx. 1 mm Fig. 8-5: Filling the cone-plate measuring system and detail of a cut-off cone tip
47 46 R ß R... ß... Outer radius of the cone Opening angle of the cone Fig. 8-6: Cone-plate measuring system according to ISO 3219 / DIN The area A in the parallel plate model corresponds with the cone area A = π R². The moving force is expressed as F = M / R Calculation of the shear stress τ from the torque M τ = 3 M 2π R 3 = G K M Torque M in [mnm], R i in [m], τ in [Pa] G K is a constant of the measuring system which depends on the cone radius. The larger the cone radius the greater is the sensitivity of the measuring system. Calculation of the shear rate The velocity v in the parallel plate model corresponds with the circumferential speed v = ω R in the rotating system. ω = 2π n / 60 where ω is the angular speed in [rad/s] und n the rotational speed in [min -1 ] Using the simplified expression tan β = β for small cone angles, the shear rate can be calculated from the rotational speed n according to the following equation: = ω β = k n where k is a conversion factor which is independent of the cone angle. At a cone angle of 1 k = 6, at an angle of 2 k = 3 and at an angle of 4 k = This means that at a constant rotational speed the shear rate is the higher the smaller the cone angle. Angles are given in [rad] or [degree], where 2π rad = 360 i.e. 1 rad = 57.3 and 1 = rad.
48 Plate-plate measuring systems The plate-plate measuring system consists of two parallel plates. It is characterised by the plate radius R and the variable distance h between the plates. The material under investigation is disposed between the two plates. DIN specifies that the plate distance h shall be much smaller than the radius of the measuring plate. A gap of between 0.3 mm and 3 mm is recommended. Plate-plate measuring devices are used if the material under investigation contains large filler particles. The gap should be determined such that it is at least five times as large as the largest particles contained in the material. The shear rate is not constant across the entire plate radius, like in the cone-plate system, but there is a relatively large shear rate range. The shear rate in the centre of the upper measuring plate is zero. The specified shear rate is always that related to the outer radius R of the measuring plate, that is the maximum shear gradient. R h R... H... Radius of the measuring plate Distance (gap) between the upper and lower measuring plate Fig. 8-7: Plate-plate measuring system The area A in the parallel plate model corresponds with the rotating area A of the upper measuring plate, where A = π R 2. The moving force is expressed as F = M / R. Calculation of the shear stress τ from the torque M τ 3 M ( R) = = G M 2π R 3 P Torque M in [mnm], R in [m], τ in [Pa] G P is a constant of the measuring system which depends on the plate radius and the distance h. The larger the plate radius the greater is the sensitivity of the measuring system.
49 48 Calculation of the shear rate The plate distance h in the parallel plate model corresponds with the distance h between the upper and lower measuring plate; the velocity v in the parallel plate model corresponds with the circumferential speed v = ω R in the rotating system. ω = 2π n / 60 where ω is the angular speed in [rad/s] und n the rotational speed in [min -1 ] In contrast to the cone-plate measuring system, the shear rate depends on the radius. It is zero in the centre of the plate (r = 0), has a maximum at the edge of the plate (r = R) and shows a linear gradient. ( R) = v h = ω R h At a constant angular speed ω or rotational speed n, if the plate distance h is increased the shear rate in the measuring gap will fall.
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