TÁMOP-4.1.2.D-12/1/KONV-2012-0008 „Szak-nyelv-tudás” - Az idegen nyelvi képzési rendszer fejlesztése a Debreceni Egyetemen

Rheology in food analysis

Dr. Sipos Péter

Debrecen, 2014

Table of contents 1. Fundamentals of rheology............................................................................................................... 3

1.1. Stress ....................................................................................................................................... 4

1.2. Deformation ............................................................................................................................ 5

1.3. Elastic rheological behaviour ................................................................................................... 7

1.4. Flow rheological behaviour ..................................................................................................... 8

1.5. Viscosity ................................................................................................................................... 9

1.6. Moduli and compliances ....................................................................................................... 10

1.7. Superpositions ....................................................................................................................... 13

2. Rheological models and classification of rheological systems ...................................................... 16

2.1. Elastic deformation ............................................................................................................... 16

2.2. Flow deformation .................................................................................................................. 18

2.3. Viscoelastic and plastoelastic deformation ........................................................................... 22

2.4. Modelling the rheological behaviour ........................................................................................ 24

2.5. Behaviour of materials under dynamic load (oscillatory testing) ......................................... 33

3. Principles of rheometry ................................................................................................................. 37

3.1. Viscometry ............................................................................................................................. 38

3.1.1. Capillary viscometers ..................................................................................................... 38

3.1.2. Falling ball viscometers ................................................................................................. 40

3.1.3. Rotational viscometers .................................................................................................. 40

3.2. Oscillatory testing ....................................................................................................................... 43

3.3. Texture analysers .................................................................................................................. 44

4. Use of rheology in food processing ............................................................................................... 48

4.1. Rheological methods in cereal analysis ...................................................................................... 48

4.2. Rheological methods in fruit and vegetable analysis ................................................................. 53

4.3. Rheological methods in the analysis of meat and meat products ........................................ 54

4.4. Rheological methods in the analysis of milk and milk products ........................................... 54

References ............................................................................................................................................. 56

1. Fundamentals of rheology The rheology is a field of physical material sciences. Its definition was invented by Professor Bingham who wrote that rheology is “the study of the deformation and flow of matter” and its short definition was accepted by the American Society of Rheology in 1929 when it was founded. The word “rheology” came from the greek “ρεσ” what means flow. By an another definition the rheology evaluates the connections between the forces acting on a material and the deformations of material what the forces cause. This approach is more general because the behaviour of material is not only flow but in several cases it is elastic of a mixed type. The rheometry is the measurement of rheological parameters of the materials.

The rheology is a part of mechanics, an intermediate scientific field between the mechanics of solid materials and hydrodynamics. It is basically a material science and tries to base constitutive equations between forces and responses, therefore they are independent from the physical appearance of the material – size, shape and volume and consider every materials as homogeneous and isotropic ones. Its methods can be classified into two groups:

• theoretical or corpuscular approach: the connections amongst forces, stresses and material response are derivable using the material science bases

• experimental or phenomenological approach: the rheological behaviour can be evaluated experimentally and the knowledge of the molecular processes is not necessary.

In the production practice the use of experimental rheometry is common. For example, the fluency of paintings is very important because they have to be thin enough to be able to glaze easily but they also have to be dense enough to remain on the surface in the required thickness to cover it. In cosmetics, the lipstick has to be soft enough to be useable properly but when it too soft it may be smeared and when it too hard it may break and can not be used. The rheological properties have especially high impacts because they show strong connections to the sensory properties: for example the chewability of a chewing gum or jelly, the crunchiness of potato chips or a biscuit or the tenderness of a roast meat can be characterized, quantified and compared to other products and technologies. On the other hand, the unit processes also require rheological basic knowledge – for example, due to the calculation of flowing speed of the material, the power requirement of mixing equipment and so on. But the rheology and rheometry are also used almost every fields of industry: polymer, metal, plastic or construction industry.

In the everyday use materials can be classified as solids and fluids (as gases and liquids). In the approach of rheology the solids can show fluid-like behaviour under special circumstances and the fluids can act like a solid material in some cases. The reason for this phenomenon is that in the everyday practice the prompt properties of materials are evaluated, but there are three influencing factors which determine the rheological properties. The first one is time, because the duration of stress strongly affects the deformation. For example, a short time extension on an elastic material (e.g. a rubber band) does not result permanent change in shape but a long-lasting load may cause irreversible deformation. The second factor is temperature; for example the viscosity and the lubricity of a motor oil strongly depends on the operating temperature and while the butter can be spread at 20°C it is flows on high temperature. The third factor is the pressure: high changes in it may result differences in rheological properties, but in the case of foodstuffs and raw materials the variability in pressure is limited, therefore its effect is considered to be negligible.

1.1. Stress The main components of rheological models are the force, stress and deformation. In rheology not the force is important, but the stress, which is the force per unit of area as it can be seen ineq1.:

𝜎 = 𝐹𝐴� 𝑁𝑚2� 1. eq.

where F is the force and A is the area on which the force uniformly applied. This means that the same force results much higher stress when the area is small between the source of force and the affected material,and it will be smaller when a large area is affected by the same force. Based on the Cauchy-principle, as the area tends to zero the F/A value tends to the σ stress value.

The stress can be uniaxial (normal), shear or bulk stress. In the case of uniaxial stress the force is applied perpendicularly to the material and the amount and direction of stress is the same in all points of the material, for example in the case of simple pressure between plates or the pulling of a rubber band. In the case of shear stress the force is applied laterally and results a heterogeneous stress in the body, for example when the butter is spread on bread. In the case of bulk stress the body is compressed from all directions simultaneously, for example as the hydrostatic pressure affects the material.

Mathematically the stress can be expressed as a matrix in real materials. When a force acts on a material the stress it results can be present as tensor and nine stress components can be separated. This separation can be seen in Figure 1., where σ presents normal (tensile) stress and τ tangential (shear) stress. In the indices the first letter shows the direction of stress component, the second one shows the normal of that plane on which affects the stress.

1. Figure: Components of stress

The stress tensor is:

σx τxy τxz σ ij = τyx σy τyz

τzx τzy σz 2. eq.

Due to the parityandequality of forces, the τyx is equal with τxy, τzx with τxz and τzy with τyz, therefore the stress status of a material can be characterized by six stress components.

1.2. Deformation Deformation is the displacement of the particles or the parts of the body from an original or reference configuration to a current one while the continuity of body is continuous. The relative deformation is called strain.

Under the influence of stress the size and shape of body changes (deformation) and new changes start in the material when the force removed. When the body returns to the original shape after the removal of force the deformation the deformation is reversible and it is called elastic deformation. When the deformation remains after the termination of stress the deformation is irreversible and it is called flow (Figure 2.). The stress results deformation on the material and this deformation also can result uniaxial, bulk and shear stress.

σzz

τzy

τzx

τ

xz

τxy

σxx

τyz

σ yy

τyx

3. Figure: Uniaxial deformation

The deformation results change in the dimensions of the body. The relative deformation (strain) can be expressed in the dimension of the body or in ratio.Figure 3.shows an uniaxial tension where the length of body changes from the original unstressed (l0) length to the l length under the influence of stress. The deformation can be expressed in the dimension of the body:

∆𝑙 = 𝑙 − 𝑙0[𝑚] 3. eq.

where l0 is the original length of the body, l is the length of body under stress and Δl is the change of length. In the case of the expression in ratio the deformation is the ratio of the change of length in the proportion to the original or the stressed length :

𝜀𝑐 = 𝑙0−𝑙𝑙0

− ∆𝑙𝑙0

[−] 4. eq.

𝜀ℎ = 𝑙0−𝑙𝑙− ∆𝑙

𝑙[−] 5. eq.

The first one is called by conventional or Cauchy deformation (ε or εC) and the second one is called natural of Hencky deformation (εH).

In the case of flow (viscous or plastic deformation) the deformation what is caused by the static load is continuously increase during the influence of load and remains after offload. It was also discussed that the rate of deformation is changes in the different planes of the material, therefore the inhomogeneous deformation is not suitable to characterize the strain. But, as it can be seen onFigure 4., the deformation can be characterized by the angle (γ) between the line perpendicular to the force

elastic deformation

flow

body under stress unstressed body after the termination of stress

2. Figure: The elasticdeformation and flow

and the endpoints of deformation lines. This case the flow strain can be expressed by the following equation:

𝑡𝑔𝛾 = ∆𝑙𝑑

[−] 6. eq.

where d is the distance between the layer on which the Δl deformation is measured and the stationary layer. In several cases the strain is small enough to substitute the tangent and the strain can be expressed as

𝛾 = ∆𝑙𝑑

[−] 7. eq.

4. Figure: Flow deformation

Rheology uses this dimensionless ratio as rheological considerations have three requirements on the bodies analysed:

1: the evaluated materials are homogeneous ones (all its points have the same properties) 2: the evaluated materials are isotropic ones (one of the selected properties of the material is uniform moving to any direction) 3: the evaluated materials can be characterised as endless ones (there are no end-effects – if the evaluated part of the material is small enough in the total body it can be considered as endless)

The connection between deformation and the stress can be immediate or time-dependent. In the first case the appearance of stress the proportional deformation can be experienced on the material while in the second case the adaptation of material requires time and the proportional deformation can be observed after a specific time.

1.3. Elastic rheological behaviour Hooke found that in the case of elastic materials the stress (σ) is proportional to the deformation (ε) and the proportionality factor depends only on the rheological properties of material. In his equation (Hooke’s law):

l Δl

γ d

𝜎 = 𝜀 ∙ 𝐸 8. eq.

where E is the elasticity or Young’s modulus [𝑃𝑎]. On the other hand, Young’s modulus shows the ratio of stress to stain. The deformation is ideally elastic when the stresses resulted by the actual external loads in the material are depend on the actual deformation and the deformation is instantaneously and fully ceased when the stress caused. The uniaxial compression-extension elastic deformations are considered to beequal amounts in all planes of the material perpendicular to the force and therefore elastic deformations are also called homogeneous deformations. When the strain is unidirectional, it is called linear deformation.

1.4. Flow rheological behaviour In the case of flow the mechanical bases are different. The force affects laterally to the body (the upper plane on Figure 5.) and it result a shear strain in the material where the velocity upper plane is maximal and the velocity (shear rate) of other planes moving to the lower plane is decrease proportionally while it reaches the value 0. The layers slip on each others, but the material has resistance again slippery which results decrease in the velocity of layers.

5. Figure: Flow deformation

In this case the deformation can be characterized by the velocity (shear rate) but it change from layer to layer, therefore the shear stress is not proportional to the shear strain. On the other hand, the lack of slipperiness can be characterized by the α angle as the v/d ratio is constant in all layers of the material. This v/d ratio is called shear rate (𝛾∙) and it is proportional to the shear stress:

𝜎 = 𝜂 ∙ 𝛾∙ 9. eq.

where the proportionality factor is the lack of slipperiness between the neighbouring layers, the resistance against flow, internal friction or viscosity [𝑃𝑎 ∙ 𝑠23T]. The Newton’s law (eq. 6.) assumes that the behaviour lack of slipperiness is proportional only to the distance to the plane of force.

As the stress can be show as tensor, it is also possible to present the deformation in a tensor and the six stress tensor components have their pairs in the deformation tensor.

As the shear rate changes from plane to plane, the flow is considered to be inhomogeneous deformation, in comparison to the homogeneous elastic compression deformations.

It was thought for a long time that the materials can be classified to elastic or flow behaviour. Weber performed experiments with loading silk threads in 1835 and he found that when a stress was

α

applied on the silk fibres they react with immediate elastic response but the remaining stress resulted flow. When the force is terminated an immediate shortening was observed and it was followed by a slow flow again while the fibres got their original length back. This experiment confirmed that there are materials which can show elastic behaviour and flow at the same time, simultaneously.

1.5. Viscosity Viscosity is also called the internal friction; it is the measure of internal resistance of fluid (gas or liquid) material against shear stress. In a common sense the higher viscosity values mean more dense fluid and the lower values are refers to thin liquids. The viscosity of ideal fluids is zero.

The viscosity is in the Newton’s equation is the dynamic viscosity (η) and it can be calculated from the eq. 9:

𝜂 = 𝜎𝛾∙

10. eq.

The fluidity of a liquid can be calculated and characterized as the reciprocal of viscosity:

Φ = 1𝜂

= 𝛾∙

𝜎 11. eq.

The kinematic viscosity (υ) is the quotient of dynamic viscosity and the density of liquid material:

υ = 𝜂𝜌

[𝑚2

𝑠] 12.eq.

where ρ is the density [gcm3-1]. Kinematic viscosity is often used in lubricanttechniques, liquids with low viscosity and in the case of foams.

In practice relative viscosity is often used, especially for low viscosity solutions. It is the quotient of the viscosity of analysed solution (η) and the viscosity of solvent (η0):

𝜂𝑟 = 𝜂𝜂0

[ ] 13.eq.

Specific viscosity is also often used in the case of solutions; it is the quotient of the difference in the viscosity of solution and solvent and the solvent:

𝜂𝑠 = 𝜂−𝜂0𝜂0

= 𝜂𝑟 − 1 [ ] 14.eq.

The viscosity can be explained by the hole theory of liquids by many authors. The liquids are compressible only in a small degree but always contain holes between the molecules. The neighbouring molecules can take place in these holes due to their heat movement what result a new hole in the material. The energy what is required to this movement is called the activation energy of viscous flow. Therefore the viscosity of Newtonian liquids can also be expressed by the theory of Eyring:

𝜂 = 𝐴ℎ𝜋𝑉𝑒3,8𝑇𝑝𝑇 15. eq.

where A is the Avogadro’s constant [1 Kmol-1], hπ is the Planck’s constant [Js], V is the molecular volume of liquid [m3 Kmol-1],Tp is the evaporation temperature of liquid [K] and T is the actual temperature of liquid [K]. This equation refers to the fact that the viscosity is strongly depends on the temperature of the material. The temperature-dependency of viscosity can be generally characterized by the Arrhenius’s law:

𝜂 = 𝐴𝑒𝑒𝑅𝑇 16. eq.

where A is a constant, E is the activation energy [Jkmol-1] and R is the universal gas constant [J kmol-1

K-1] and T is the temperature [K]. For other materials other equations are also used and give better result. For example the shift factor and WLF equation is also used for solid and diluted plastics and glasses:

𝑙𝑜𝑔 𝜂𝑇𝜂𝑇0

= −8,86(𝑇−𝑇0)101,6+𝑇−𝑇0

17. eq.

where T is the temperature of observation and T0 is the reference temperature. The increase in temperature decreases the viscosity because kinematic energy of molecules increases and the difference between kinematic and viscous forces decreases.

On the other hand, the size of molecules is against their movement; the increase in molecule size strongly increases the viscosity:

𝑙𝑛𝜂 = 𝐵 + 3,4𝑙𝑔𝑀 18. eq.

where B is a constant and M is the molecular weight [kgmol-1].

The increasing pressure increases the viscosity and it can be experienced during the processing of foodstuffs (for example, due to the thermal expansion). The connection between viscosity and pressure can generally characterized by the following equation:

𝜂 = 𝜂0𝑒𝑐𝑝 19. eq.

where η0 is the viscosity at atmospheric pressure [Pa s], c is a factor and p is the pressure [Pa].

1.6. Moduli and compliances By the definition moduli show that how much stress is required to make one unit of deformation. Its unit is the same as the unit of stress [Pa]. Their reciprocal are the compliances and they present that how much deformation can be experienced by one unit of stress. In rheology in the case of elastic deformation the Young’s or tensile modulus (E) is the quotient of strain and stress and its reciprocal is D, namely the tensile compliance:

𝐸 = 𝜎𝜀[Pa] 20.eq.

𝐷 = 𝜀𝜎

[ 1𝑃𝑎

] 21.eq.

In the case of viscous materials the shear modulus (D) presents the rate of stress necessary for an unit of deformation and it is also called the modulus of rigidity. Its reciprocal is the shear compliance and it shows that how much deformation occurs by one unit of stress:

𝐺 = 𝜎𝜀[Pa] 22.eq.

𝐽 = 𝜀𝜎

[ 1𝑃𝑎

] 23.eq.

The bulk modulus (K) is a parameter characterizes the behaviour of a material against uniform (bulk) stress. By the definition it is the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. Its reciprocal is the bulk compliance

𝐾 = −𝑉 ∆𝑝∆𝑉

= 𝜌 ∆𝑝∆𝜌

[Pa] 24.eq.

𝐵 = 1𝐾

[ 1𝑃𝑎

] 25.eq.

where p is the pressure [Pa], V is the volume [m3] and ρ is the density [kg m-3]

When an unit of stress results large deformation the modulus is small and the compliance is large, so the more hard and resistant materials have large moduli (as their compliances, sensitivity to stress are lower).

The different moduli and tensile values are strongly related to each other. The general correlation is described by the following equations:

𝐷 = 𝐽3

+ 𝐵9

26. eq.

and

𝐸 = � 13𝑔

+ 19𝐾�−1

27. eq.

The material used in food science are generally incompressible, therefore the bulk stress and bulk modulus is negligible. In practice the following equation is also valid:

𝐸 = 2𝐺(1 + 𝜇) 28. eq.

where

𝜇 = 12�1 − 1

𝜀∆𝑉𝑉� 29. eq.

μ is the Poission‘s ratio. It can be seen that small (or negligible) change in volume result that the Poission’s ratio is very small and it may be omitted from the relation:

𝐸 = 3𝐺 30. eq.

𝐽 = 3𝐷 31. eq.

1.7. The effect of time and temperature on rheological behaviour The time and temperature dependency of rheological behaviour can be seen in the change of moduli by the change of these factors. Figure 6.illustrates the changes of Young’s modulus by temperature. It can be seen that very short stress results large E value and the material shows very small deformation. This region is the glassy region where the material acts like a rigid material with a very high E value (E1) and the increasing temperature has negligible or a very small effect on it only. By the increase of temperature the modulus show rapid decrease (this is the transitional stage). In the case of plastics it is a very short temperature region (5-10°C). Later the E stabilizes again. It is the elastic or rubbery region of the curve while the material meets the Hooke’s law and the Young’s modulus is constant (E2).Higher temperature results decrease again in modulus and the material starts to flow or melt (flow region).

6. Figure: Temperature dependence of Young's modulus of a viscoelastic material

Experiments proved that the effect of temperature on the change of moduli follows this tendency and the same stages can be experienced. The time dependence on the behaviour of a viscoelastic material can be seen on Figure 7. A very short stress result large E value while the longer strains increase the rate of deformation. Therefore short stress result solid-like behaviour and long stress fluid-like behaviour in the case of the same material.

7. Figure: Time dependence of the Young’s modulus of a viscoelastic material

For define the state of matter in rheology Deborah number is used:

𝐷𝑒 = 𝜏𝑡 32. eq.

whereτ is the characteristic time or Maxwell’s relaxation time) and t is the observation time scale . The characteristic time of Hooke’s elastic bodies is practically unlimited resulting high Deborah number and the rigid and elastic materials have high values. Lower values mean more viscous behaviour. The Deborah number is 1-10 s in the case of plastics and 10-12 in the case of water.

1.8. Superpositions In normal case not only one force and stress acts on the material but more. The Boltzmann’s superposition principle simply defines the connections between simultaneous stresses and deformations. When 2 stresses act on the material the arising stress can be calculated as the sum of stresses:

𝜎𝑎𝑐𝑡𝑢𝑎𝑙 = 𝜎1 + 𝜎2 33. eq.

and the shear strains what the stresses cause are also can be summarized:

𝛾(𝜎1) + 𝛾(𝜎2) = 𝛾(𝜎1 + 𝜎2) 34. eq.

and it is also true for the shear rates:

𝛾 .(𝜎1) + 𝛾 .(𝜎2) = 𝛾 .(𝜎1 + 𝜎2) 35. eq.

The principle is illustrated on Fig. 5.

logE

logt

0 -

-4 -

-2 -

-6 -

-2 0 2 4 6 8 10 12

The second superposition principle used in rheology is the time-temperature superposition. As the changes of behaviour of the materials are the same by the change of time and temperature these factors can substitute each other and a stress on higher temperature for a shorter time results the same deformation as an another stress result on the material for a longer time on lower temperature. This principle is a very useful one in rheometry because the effect of long stress can be substituted and therefore modelled by the application of higher testing temperature.

Due to the similarities of the rheological behaviour under the influences of changing time and temperature the effect two factors can be compared with each other, thus the changes of Young’s modulus by the increasing temperature allows to calculate the effect of longer stress time. The evaluation of rheological properties of a material on different temperatures for predefined time scales (from a few seconds to hours) result partial curves which make possible to create a master curve which present the change of elasticity modulus by time on a specific temperature. An example for the construction of master curve can be seen at http://www.open.edu/openlearn/ocw/pluginfile.php/75528/mod_oucontent/oucontent/640/none/none/t838_1_047i.jpg.

The connection also can be expressed in equations:

𝐸(𝑇1; 𝑡) = 𝐸(𝑇2; 𝑡𝑎𝑇

) 36. eq.

where T1 and T2 are the different observation temperatures, t is the time scale of observation and aT is the shift factor. Later it was found that the increase in temperature often results change in density which makes the correction of eq. 36.necessary:

t

8. Figure: The Boltzmann’s superpositionprinciple

𝐸 �𝑇; 𝑡𝑎𝑇� = 𝐸(𝑇0; 𝑡) 𝜌(𝑇)𝑇

𝜌(𝑇0)𝑇0 37. eq.

where ρ(T0) and ρ(T) is the density of material on reference and observation temperature. The time-temperature superposition principle also can applied on other rheological parameters such as moduli and compliances. The shift factor can be determined many ways. By definition aT is the quotient of load time scales which result the same rheological effects:

𝑎𝑇 = 𝑡𝑇𝑡𝑇0

38. eq.

wheretT is an arbitrary temperature and t0 is the reference temperature. Practically the times are replaceable with viscosity values:

𝑎𝑇 = 𝜂𝑇𝜂𝑇0

39. eq.

The shift factor can be calculated by the Arrhenius’s law and the WLF (Williams-Landel-Ferry) equation. In the first case the changes in rheological behaviour is calculated by the dependence of chemical reaction speed on temperature while in the second case the connection has a molecular based approach and the calculation is based on the free volume theory.

2. Rheological models and classification of rheological systems The three basic rheological behaviour types are the elastic, viscous and plastic ones.All types can be classified further to ideal and not ideal groups and linear and non-linear groups and the combinations of them. The system of rheological behaviours is summarized in Figure 9.

9. Figure: Classification of the rheological systems

2.1. Elastic deformation Based on the classification it can be seen that there three base behaviour types and their combinations. The base rheological systems are the elastic, viscous and plastic and the combinations are the viscoelastic and plastoelastic behaviour type.

The elastic deformation means that the deformation is proportional to the actual stress and when the stress is terminated the material returns to its original shape. These processes can be seen in Figure 10.It can be seen that the stress results immediate deformation, therefore this is the ideal elastic rheological response.

deformation

elastic flow

ideal non-ideal

linear non-linear reversible irreversible

plastic viscous

linear (Newtonian)

non- Newtonian

Bingham-

system non-linear plastic

plastoelastic viscoelastic

10. Figure: Time-stress and time-strain diagrams of ideal elastic rheological systems

The connection between stress and deformation can be linear (linear ideal elastic behaviour) or non-linear, where the increasing strain results a more or less intensive increase in stress (Figure 11.). The non-linear ideal elastic behaviour results immediate cease of stress when the load is terminated.

11. Figure: Stress-strain diagrams of the linear and non-linear ideal elastic rheological systems

In the case of non-ideal elastic systems the connection between stress and strain is not instantaneous, but the material shows increase in strain by time under the effect of constant load. The non-ideal elasticity can be linear and non-linear, also and reversible or irreversible. In the case of reversible non-linear elastic systems the by unload of stress the deformation will be ceased and the material returns to its original shape (Figure 12.). In the case of irreversible non-linear elastic

systemsthe behaviour of the material is not completely elastic but it also shows flow - viscous or plastic behaviour.

12. Figure: Time-stress, time-strain and stress-strain diagrams of non-ideal reversible elastic rheological systems

2.2. Flow deformation The flow rheological behaviour is classified to viscous and plastic kinds. In the case of viscous materials the smallest stress result immediate and irreversible deformation, while in the case of plastic materials there are no permanent deformation below the yield stress. The connection between shear stress and shear rate is characterized practically by the general equation:

𝜎 = 𝐾𝛾 .𝑛 40. eq.

where K is the consistency and n is the shear rate index.

The Newtonian fluids show ideally viscous behaviour where the shear stress is linearly proportionally to shear rate and the coefficient is the viscosity.In the general equation of flow behaviour (40. eq.) the n is 1 (linear connection) and the K is the viscosity. In the case of Newtonian behaviour the stress-shear rate reaction is immediate, the stress results continuous deformation and after the termination of load the last deformation stage remains permanently (Figure 13.).

13. Figure: Time-stress, time-strain, shearrate-stress diagrams and shear rate-viscosuty curves of ideally viscous or Newtonian rheological systems

The non-linear Newtonian viscous behaviour means that the connection between shear stress and shear rate is not linear (14. Figure) and the deformation is permanent after the unload. This means practically that the viscosity of material changes with the deformation speed.

14. Figure: Shear rate-shear stress diagrams of non-linear viscous rheological systems

If the n is lower than 1 in the general equation of flow behaviour (40. eq.) the rheological behaviour is called shear thinning materials. In this case the increase of shear stress caused by the increasing shear rate shows decreasing tendency (upper curve on Figure 14.) and the viscosity of this material decreases by the increasing shear rate (Figure 15.), therefore the viscosity of these material is not constant, changes continuously. It is called apparent viscosity. The connection between viscosity and shear rate can be presented on logarithmic scale as it results linear connection on higher shear rate regions.

15. Figure: Influence of shear rate on the viscosity of a shear thinning material

When the viscosity of a shear thinning material is analysed from the very small shear rate region to the very high three sections can be separated. The slow shear rate region results a nearly constant

viscosity value, the same can be experienced in high rate regions too and the shear thinning behaviour can be experienced in the middle region. Therefore, the viscosity values of these materials are independent from deformation in the first and the third sections: these are called the first and second Newtonian regions (Figure 16.). The endpoint of the first Newtonian region can be marked by η0 viscosity value as the beginning of the shear thinning region and the start point of the second Newtonian region can be characterized by η1 viscosity value. In this case the 40. eq. is valid only in the shear thinning region.

16. Figure: The shear rate-viscosity curve of shear thinning material

The reason for shear thinning behaviour explains the reasons for the presence of three regions too. The fluids contains particles which resistant against the strain. These can be solid materials in the case of suspensions or dissolved molecules in the case of solutions and their resistance depends on their spatialarrangement. As an effect of the increasing shear rate these particles start to change their shapes – for example fibrillar particles start to orientate or elongate, spherical particles start to order or deform. The very low shear rate makes possible to the liquid parts to flow between these particles without having influence on them but as the strain increases the realignment become more advanced. The aligned structure result less resistance against flow, therefore decrease in viscosity until the point when it reaches the minimum value. On the other hand, when the strain ceases the spatialarrangements of molecules start to arrange back to the original conformation and the material regain its original viscosity within more or less time.

When the n is lower than 1 in the general equation of flow behaviour (40. eq.) the rheological behaviour of the material is called shear thickening.In this case the shear stress increasingly increases by the shear rate while the increase of viscosity of the material shows decreasing tendency (Figure 17.). The change of viscosity is reversible similarly to the shear thinning materials. The shear thickening materials are rare in the nature but they have significant importance in food industry as the structure of special product requires this kind of behaviour, for example different starch suspensions.

17. Figure: Connections between shear rate, shear stress and viscosity of a shear thickening material

The plastic rheological behaviour means that the material behave as rigid material until a yield value of shear (yield stress) and flow as Newtonian liquids above this value.The 40.eq. is supplemented by the yield stress (σ0):

𝜎 = 𝜎0 + 𝐾𝛾 .𝑛 41. eq.

In the most cases the yield stress depends on the temperature and above a specific temperature the material acts like a viscous material. The typical shear rate - shear stress curve of a linear viscous rheological material (Bingham system) can be seen in Figure 18.

18. Figure: The shear rate - shear stress curve of a linear viscous rheological material

When the stress response of material is not linear to the shear rate above the yield stress the plastic material have non-Newtonian behaviour and behave like a shear thinning material it is called a pseudoplastic rheological or generalized Bingham system.

The most shear thinning and shear thickening materials (as non-Newtonian materials) do not show difference in viscosity value when the shear rate increases or decreases: both direction of change result the same viscosity values. In some cases it van be observed that the viscosity values are different during the increase and the decrease of shear rate at the same lead level. These are the third basic group of viscous behaviour groups: these materials show hysteresis and the two general

types are the thixotropic and rheopexic behaviour. These are reversible rheological behaviour kinds but the realignment requires time, this is why these groups are called time-dependent behaviour groups.

In the case of thixotropic viscous systems the viscosity value correspond to a selected shear rate value is higher when the shear rate increases and lower when the shear rate decreases. Generally thixotropic behaviour was experienced on sol-gel systems and in the case of small solid particles which bond thick liquid layers. The rheopexic behaviour is the opposite of thixotropical: rheopexic systems show higher viscosity with the decreasing shear rate (Figure 19.). The difference in the viscosity values in the ascending and the descending part of curve depends on the range of shear rate, the shear force and the time of shear.

19. Figure: The thixotropic and rheoplexic behaviour

2.3. Viscoelastic and plastoelastic deformation The real materials rarely show pure elastic or flow behaviour. Generally the materials show elastic behaviour under the influence of small or relative small stresses and above a specific value they start to flow and then they are unloaded a partial rearrangement can be observed. In these cases the changes and rearrangement of the material is not instantaneous but show time dependency. The elastic limit and the type of change strongly depend on those parameters which generally influences the rheological behaviour: temperature and pressure.

The viscoelastic and plastoelastic rheological behaviour means that the elastic and viscous or plastic deformation can be observed in the material in the same time. References says that the viscoelastic behaviour can be proved when the material shows delayed elasticity, i.e. the termination of deformation result partial or full realignment bit it is not instantaneous but has a decreasing speed. On the other hand, viscoelastic systems also show the phenomenon of stress relaxation, what means that the viscoelastic material loaded by a constant stress shows decrease in stress by deformation time. It implies that the viscoelastic systems can be characterized by the shear compliance (J) and relaxation modulus (G), but as the viscoelastic behaviour time dependent, these values are also time dependent ones. The time dependent shear compliance is the quotient of the actual deformation

and the shear stress (42. eq.) and the time dependent relaxation modulus, what is the quotient of the actual shear stress and the deformation (43. eq.).

𝐽(𝑡) = 𝛾(𝑡)𝜏0

42. eq.

𝐺(𝑡) = 𝜏(𝑡)𝛾0

43. eq.

The evaluation of viscoelastic rheological behaviour can be done by the evaluation if strain-time curve of the material under the effect of constant load and later unload (Figure 20.). When the stress is applied on the material an immediate and increasing deformation can be observed but the rate of increase is decreases and it tends to a limit value – this is the maximum deformation what the stress can induce. In the t1 time the stress is terminated and the deformation starts to decrease but the speed of realignment shows continuously decreasing tendency until the material reaches the new unstressed shape with a permanent deformation. The first, loaded part of the deformation-time curve of a viscoelastic material is the creep region and the second, unloaded part is the relaxation region. The longer period of stress-effect evaluation results more parts on the curve; the creep region can be divided into three parts: the first one is the previously presented one, the second part is an apparently linear part (or when the shear strain is analysed by time, a constant shear strain is experienced) and the third part is an increasing part when the speed of deformation start to increase again. In practice the first and the second part is evaluated by rheometry.

20. Figure: Typical creep and relaxation curve of a viscoelastic material

In the creep region a time-dependent tensile and shear can be experienced and their effect can be characterized by the time dependent tensile strain and time dependent shear strain: 𝜀(𝑡) =𝐷(𝑡)𝜎and 𝛾(𝑡) = 𝐽(𝑡)𝜎. In the recovery region the time dependent tensile relaxation modulus and

time dependent shear relaxation modulus: 𝐸(𝑡) = 𝜎(𝑡)𝜀

and 𝐺(𝑡) = 𝜎(𝑡)𝛾

.

Three fields of the analysis of viscoelastic behaviour are common:

• the evaluation of time dependent deformation under constant stress • the evaluation of relaxation after unloading the stress • the evaluation of stress and its relaxation under a constant deformation

2.4. Modelling the rheological behaviour The modelling of the rheological behaviour of different materials are interesting topics for researchers for a long time. The creation of models helps to visualize the systems and understand the real mechanisms in the material.

The models can be made by two ways: mechanic and electronic analogies can be used in the creation. The basic elements of models are the followings:

rheological parameter mechanical analog electronic analog stress mechanical stress (load) power source deformation movement charge shear rate speed of movement amperage ideally elastic material or deformation

spring

capacitor

ideally viscous material or deformation

piston moving in a pot of fluid

resistor

ideally plastic material or deformation

body frictioning on a surface

resistor with limit opening amperage

serial connection of elements deformations are summarized stresses are summarized parallel connection of elements stresses are summarized deformations are summarized

The most references use the mechanical modelling to present and explain the simplest connections of rheological properties. The models use the serial and/or parallel linking the mechanical elements and present the changes of stress and strain by time.

The ideally elastic deformation can be characterized by a single spring which is loaded by a constant load (Figure 21.). In this case the deformation is immediate when the load is started and constant during the influence of constant load. The stress resulted by the load is proportional to the load and also constant. When the material is instantaneously become unloaded the stress is instantaneously terminated also and the strain proportional to stress is also instantaneously ceased. The Young’s moduli can be substituted by the spring constant.

21. Figure: Mechanical model and changes of stress and strain of an ideally elastic material (Hooke's body)

The mechanical model of the Newtonian fluids or ideally linearly viscous materials is a piston moving in a pot of fluid to an infinite distance (Figure 22.). In this case mechanical load starts to move the piston in the liquid and the speed of movement is determined by the viscosity of fluid in the dashpot but proportional to the load. The appearance of constant load result a constant stress in the material and the constant speed motion can be interpreted as the constant shear strain. When the load instantaneously terminated the stress is ceased again and the position of piston is permanent (as well as the deformation) because no more stress to maintain it and the liquid material blocks the further movement by inertia.

22. Figure: Mechanical model and changes of stress and strain of an ideally viscous material (Newtonian fluid)

The mechanical model of an ideally linearly plastic system is a Saint-Venants body which is a body which is adheres to a surface. When the load is smaller than the friction force deformation cannot be experienced in the material, but a stress proportional to the load can be measured. When the load is higher than the friction force the same diagrams can be experienced as in the case of Newtonian

liquids – the deformation is proportional to the stress. When the material is unloaded the final deformation remains permanently.

These three elements are enough to model the difficult rheological systems by their parallel or serial connection and their combinations. The most general models are the followings:

• viscoelastic models: elastic-viscous systems (Maxwell and Kelvin-Voigt models) • plastoelastic model: elastic-plastic system • viscoplastic model (Svedov-Bingham model)

The Maxwell model is a serially connected elastic material which can be characterized by the Young’s modulus (E) and viscous material which can be characterized by the viscosity (η) (Figure 23.). The changes of stress and deformation when the effect of a constant load is analysed by time can be seen on Figure 24. When the load is applied on the body (from t1 time – creep region) the same stress can be measured in both the elastic and viscous component (𝜎0 = 𝜎𝑒 = 𝜎𝑣) (symbolized by the first red vertical line on the stress curve), but their response in deformation is different. An instantaneous elastic deformation can be experienced in the elastic part (the first green vertical line on the deformation curve) and the same time the continuous deformation of viscous part is starting (the rising green line). The slope of the curve is depending on the viscosity. When the model is unloaded (from t2 time – relaxation region) the elastic component returns to the original position (reversible part of deformation, symbolized by the second green vertical line) and the deformation of viscous element remains permanently (irreversible part of deformation, presented by the last horizontal green line).

23. Figure: The Maxwell model

24. Figure: The time-stress and time-deformation curves of the Maxwell model

In the creep region the deformation is time dependent due to the continuous deformation of viscous part and the actual total deformation is the sum of the deformation of elastic and viscous parts

(𝛾(𝑡) = 𝛾𝑒 + 𝛾(𝑡)𝑣R). In the relaxation region the elastic deformation ceased and the shape of this part is returned to the original unloaded shape and the actual deformation is only the viscous deformation (𝛾(𝑡) = 𝛾(𝑡)𝑣R). The actual deformation in any time can be expressed by the following equation:

𝛾(𝑡)0 = 𝜎𝑇𝑀𝜂∙ �1 + 1

𝑇𝑀𝑡� 44. eq.

where t is the time of observation and TM is the relaxation time. By definition Maxwell’s relaxation time is the quotient of viscosity and elasticity modulus:

𝑇𝑀 = 𝜂𝐸

45. eq.

The second possible use of Maxwell model is to analyse the stress necessary to maintain a constant deformation (γ=a). This can be modelled when the body of Maxwell model is stretched instantaneously to a constant length. The elastic element is also instantaneously elongated to the length what is proportional to the stress resulted by the deformation and in the first moment of stresses status the viscous deformation is zero but it starts as the stress has influence on it. The deformation of viscous component decreases the stress in the material (it start to tends to zero) and the decrease in stress results decrease in the deformation of the elastic component. At the end of the relaxation period the value of stress is zero what induces that the all the deformation comes from the deformation of piston as viscous part and the spring as the elastic part returned to its original unloaded shape. The actual stress in any t time can be expressed as

𝑙𝑛𝜎𝑡 = 𝑙𝑛𝜎0 −𝐸𝜂𝑡 46. eq.

where𝜎0 is the stress at unloading, E is the Young’s modulus of elastic component and η is the viscosity of viscous component. The introduction of Maxwell’s relaxation time in the equation result a simpler equation:

𝜎𝑡 = 𝜎0𝑒−𝑡𝑇𝑀 47. eq.

This equation can define the unit of relaxation time: it is the time required to that the stress decrease to the eth part its original value (0,369), therefore the stress in materials act like Maxwell model decreases exponentially (Figure 25.).

25. Figure: Explanation of relaxation time

The Kelvin-Voigt modelis a parallel connected elastic material which can be characterized by the Young’s modulus (E) and viscous material which can be characterized by the viscosity (η) (Figure 26.). The changes of stress and deformation when the effect of a constant load is analysed by time can be seen on Figure 27. When the load is applied on the body (from t1 time – creep region) the stress is divided by the elastic and viscous parts of the model equally (as the two connecting rods under and above the rheological elements are parallel to each other continuously) (𝜎0 = 𝜎𝑒 + 𝜎𝑣). Therefore the deformation on both sides is the same (𝛾0 = 𝛾𝑒 = 𝛾𝑣) and while the elastic part would make immediate deformation possible, the viscous part holds it back. As the model deforms the stress decreases because the elastic part absorbs the proportional part and the deformation speed of viscous part decreases – this is why the deformation curve (green curve on Figure 27.) shows increase with decreasing tendency until it reaches the maximum deformation what is distributed to the elastic part. In the moment of total deformation the 𝛾0 = 𝛾𝑒 = 𝛾𝑣 equality is also valid but while the stress on elastic side is equal to the total stress the stress on the viscous part is zero (𝜎0 =𝜎𝑒 𝑎𝑛𝑑 𝜎𝑣=0).

26. Figure: The Kelvin-Voight model

27. Figure: The time-stress and time-deformation curves of the Kelvin-Voight model

The reaction is similar in the relaxation stage when the stress is terminated. The elastic part initiate immediate realignment to the original position but the viscous part makes it slower by its viscosity. The relaxation time is influenced again by the E and η values. The total stress in the model can be expressed as

𝜎0 = 𝐸𝛾 + 𝜂 𝑑𝛾𝑑𝜂

48. eq.

In this case the η/E quotient is called delay time. The deformation can be calculated:

𝛾 = 𝜎𝐸�1 − 𝑒

−𝑡𝑇𝑘� 49. eq.

whereTk is the delay time.

The explanation of delay time is similar to the explanation of relaxation time: it is the 1-1𝑒

P

th part of

the time what is necessary for the deformation to reach the maximum deformation, or necessary to the decrease of the deformation to the eth part after unload (Figure 28.). Although the relaxation and delay time are similar parameters based on the differential theoretical background in rheology they are managed separately but in practice they are not differentiated and marked T or τ (see 32. eq.)

28. Figure: Explanation of relaxation time

Based on the properties of rheological behaviour (deformation time is infinite for the ideally elastic part is and zero for the elastic part) the time of alignment and realignment of these models are theoretically infinite.

The plastoelastic model is a serially or parallel connected elastic material (spring) which can be characterized by the Young’s modulus (E) and plastic material (a body frictioning on a surface) which can be characterized by the viscosity (η) and yield stress (σ0).When the elements are connected serial (Figure 29a.) the stress is the same again on the two elements and it is also equal to the total stress. When a stress less than the yield stress is applied on it the elastic part shows proportional deformation and the plastic part remains undeformed, but when the stress is higher than the yield stress the plastic part start to deform and the deformation of elastic part will be permanent (Figure 30.). When the stress ceased the elastic part shows immediate realignment to the original shape and the deformation of plastic part will be permanent. Then the connection of elements are parallel, the plastic part do not allow the elastic part to deform under the yield stress, but when the stress reaches the yield value the deformation and the realignment after unload will follow the same tendency as it was presented in the case of Kelvin-Voigt model (Figure 27.).

a) b)

29. Figure: Plastoelastic models

30. Figure: Deformation-stress diagram of a serial plastoelastic model

The Svedov-Bingham model models the viscoplastic system when a viscous system characterized by η1 viscosity and a plastic system characterized by η2 viscosity and σ0 yield stress is serially connected. In this case while the stress is less than the yield stress the material acts like a rigid material and only then the stress is higher than the yield stress starts to deform.

The last generalized model is the Burgers model which is a serially connected Maxwell and Kelvin-Voigt model (Figure 31.).

31. Figure: Burger model

Due to the serial connection the stress in the same on the elastic, the viscous and the Kelvin-Voigt model parts of the model (𝜎0 = 𝜎𝑒 = 𝜎𝑣 = 𝜎𝐾𝑉). The total deformation of the model under a specific load can be calculated by the 50. eq., e.g. the total deformation is the sum of the deformations of the three elements. The deformation-time diagram of the creep region can be seen in Figure 32.

𝛾 = 𝜎0𝐸1

+ 𝜎0𝜂1

(𝑡) + 𝜎0𝐸2

(1 − 𝑒�𝐸2𝜂2(𝑡)�

) 50. eq.

In the t1 time the load result an immediate deformation on the elastic element (E1) and the viscous element (η1) and the Kelvin-Voigt element (E2-η2) start a continuous deformation. Therefore in the t1

time the deformation is 𝜎0𝐸1

and from t1 to t2 time it is supplemented with the 𝜎0𝜂1

(𝑡)linear deformation

of η1 viscous element and the 𝜎0𝐸2

(1 − 𝑒�𝐸2𝜂2(𝑡)�

) deformation delayed in time of the Kelvin-Voigt

element (E2-η2). When the stress is terminated in t2 time, an immediate realignment of elastic

element decreases the deformation by 𝜎0𝐸1

and the realignment of Kelvin-Voigt element start resulting

a 𝜎0𝐸2

(1 − 𝑒�𝐸2𝜂2(𝑡)�

)relaxation. The deformation of body tends to the irreversible deformation of

viscous element (𝜎0𝜂1

(𝑡)(32. Figure) but due to the properties of the viscous part of Kelvin-Voigt

element (η2) it will be an infinite movement.

32. Figure: The time-stress and time-deformation curves of the Burgers model

Several other models were also created to characterize the different rheological behaviour. The Bingham model is a parallel viscous and plastic element and they are serially connecting to an elastic element. The Svedov model contains a serially connected viscous and elastic part, which is parallel to a plastic element and they are serially connected to a second elastic part. The Maxwell-Thompson model is a Kelvin-Voigt model and an elastic element connected serially. Macsihin and Macsihin (1987) present in their work the Gupta and Tscheuschner’s finding wherewith they model the response of fresh bread cutting process using the Shofield-Scott-Blair modell (Figure 33.) which is an serially connected elastic, plastic and a plastoelastic element. When the bread is cut, first an elastic movement can be described an immediate elastic (E1) and delayed plastoelastic (E2, η2) deformation. When the stress reaches the yield stress of dough the plastic reaction starts (η1) and irreversible deformation can be found on the bread and the cut is started.

33. Figure: Shofield-Scott-Blair modell

To characterize the real material several generalized model are developed. For example, the generalized Maxwell model contains n piece Maxwell element, the generalized Kelvin-Voigt model contains n piece Kelvin-Voigt element. The advantages of the use of these models are they are easy to understand the behaviour of material, easy to formulate the demand and no difficult theroretical differential equations are required to build. On the other hand, even the relatively homogeneous materials are hard or impossible to describe by a model with only a few elements but due to the increase in the number of components in model it loses its advance of simplicity. However, the electronic model (not discussed in this book in details) can help rheologists to build difficult models

using modern informatics tools what is a great opportunity, because it easy and costless to use to define the demands on a material, to estimate the behaviour of any products in different conditions, such as high or low temperature and on different time scales.

34. Figure: The generalized Kelvin-Voigt model

2.5. Behaviour of materials under dynamic load (oscillatory testing) The rheological evaluations can be classified into static and dynamic tests. In the case of static tests the stress or the deformation is constant and the other parameter is investigated (see the graphs presented in the description of base rheological systems). The advantages of these methods are that they are relatively easy to perform and give relevant information, but they can be accepted only in predefined conditions. In the case of dynamic tests one parameter is change during the test (stress, deformation, temperature, pressure, etc.) and helps to gain further parameters in comparison to the static tests.

The principle is the dynamic test is that a periodically changing dynamic stress or deformation is applied on one side of the tested material and the deformation or stress is measured continuously on the opposite side. The stressed side of material is connected to a rotating eccentricdisc and the rotational movement is transformed to alternating motion. The scheme of measurement and the stress-time and deformation-time can be seen on Figure 35. The time dependent sinusoidal stress generated on the material result a deformation response in the material and the result of this stress is measured on the opposite side of equipment as deformation. The phase angle shift between the stress and deformation give information about the kind of viscoelasticity.

The basic requirements on the dynamic tests are that the analysed material has to be homogeneous and isotropic, its inertia has to be negligible and the phase angle between the stress and deformation is independent from the amplitude and depends only on the behaviour of the material.

In the case of ideally elastic materials the deformation response is in the same phase to the stress. The placement of rod on the disc with different radius results change in the amplitude of alternate movement, therefore change in shear deformation and shear rate.

35. Figure: The principle of dynamic test of an elastic material and its stress-time and deformation-time curve

In the case of the test of ideally elastic body (Figure 35.) the actual stress under periodical stress load is:

𝝈 = 𝝈𝟎 ∙ 𝐜𝐨𝐬 𝝎 ∙ 𝒕 51. eq.

where σ0 is the maximum stress, what can be measured on the disc tangentially and ω is the angular velocity, what determines that what ratio of maximum stress is applied on the material in t time. The actual deformation is proportional to the stress and in this case it follows the stress without phase angleshift:

𝜸 = 𝝈𝑬

= 𝝈𝟎𝒄𝒐𝒔𝝎𝒕𝑬

52. eq.

The scheme of dynamic test on viscous material can be seen on Figure 36. The actual stress can be calculated similarly to the one for the elastic material (51. eq.), but the deformation follows the stress by 90°phase angle shift and the actual deformation can be calculated as:

𝜸 = 𝝈𝟎𝒔𝒊𝒏𝝎𝒕𝝎𝜼

53. eq.

ω

36. Figure: The principle of dynamic test of a viscous material and its stress-time and deformation-time curve

The behaviour of viscoelastic materials shows elastic and viscous properties at the same time, therefore the common deformation response of viscoelastic material to the stress is:

𝜸 = 𝜸𝟎𝒄𝒐𝒔𝜹𝒄𝒐𝒔𝝎𝒕 + 𝜸𝟎𝒔𝒊𝒏𝜹𝒔𝒊𝒏𝝎𝒕 54. eq.

whereδ is the phase angle shift and𝜂0𝑐𝑜𝑠𝛿𝑐𝑜𝑠𝜔𝑡 is the deformation component which is in the same phase angle as the stress (elastic component) and 𝜂0𝑠𝑖𝑛𝛿𝑠𝑖𝑛𝜔𝑡 is the one which shows δ=90° phase angle shift to the stress (viscous component). In rheology, the deformation which is in the same phase as the stress is called rear compliance or shear storage compliance and its symbol is J’ and the one which is in 90°phase angle shift is called imaginary compliance or shear loss compliance and its symbol is J’’. Therefore the actual deformation for viscoelastic materials can be described as:

𝜸 = 𝝈𝟎(𝑱′𝒄𝒐𝒔𝝎𝒕 + 𝑱′′𝒔𝒊𝒏𝝎𝒕) 55.eq.

The quotient of storage (real) and compliance loss (imaginary) is the tangent of phase angle shift.

Similarly, the complex stress can be expressed as σ0eiωt, where 𝑖 = √−1and the complex strain amplitude can be expressed as γ0e-iδ. J* is the complex shear compliance what is the quotient of complex strain amplitude and the complex shear compliance:

𝐽∗ = γ0e−iδ

𝜎0= 𝛾0

𝜎0(𝑐𝑜𝑠𝛿 − 𝑖 sin𝛿) = 𝐽′ − 𝑖𝐽" 56. eq.

When a viscoelastic body is under the effect of periodical deformation, the moduli values can be expressed similarly to the compliance values. The G’ storage modulus is the quotient of the amplitude of the stress in phase with the deformation (σ0cosδ) to the amplitude of the strain (γ0); the G” loss modulus is the quotient of the amplitude of the stress component in 90° phase shift to the deformation (σ0cosδ) to the amplitude of the strain (γ0) and the complex modulus is the ratio of amplitude of complex shear stress and complex shear deformation:

𝐺∗ = σ0eiδ

𝛾0= 𝛿0

𝛾0(𝑐𝑜𝑠𝛿 + 𝑖 sin𝛿) = 𝐺′ + 𝑖𝐺" 57. eq.

The connection between the complex modulus and compliance is similar to the ones presented earlier:

𝐺∗ = 1𝐽∗

= 1𝐽′−𝑖𝐽"

58. eq.

Similarly to the moduli and compliances the viscosity also can be expressed in complex, storage and loss forms. The complex viscosity is the ratio of the amplitude of complex shear stress and the amplitude of complex strain rate. As the shear rate is equal to iωt, the complex viscosity is:

𝜂∗ = 𝜎0𝑖𝜔𝛾0𝑒−𝑖𝛿

= 𝜎0𝑒𝑖𝛿

𝛾0𝑖𝜔= 𝐺∗

𝑖𝜔 59. eq.

The complex viscosity has two components: η’ is the storage viscosity what is the ratio of loss modulus and angular velocity and loss viscosity what is the ratio of storage modulus and angular velocity. In rheology the storage viscosity is actually the dynamic viscosity and the loss viscosity is also called as the out-of-phase component of complex viscosity. The connection between the viscosity values can be expressed similarly to the moduli and compliances, for example

𝜼∗ = 𝜼′ − 𝒊𝜼" 60. eq.

or

|𝜼∗| = �𝜼′𝟐 + 𝜼"𝟐 61. eq.

and the phase angle shift is:

𝒕𝒈𝜹 = 𝜼′𝜼"

62. eq.

The connection between dynamic and complex viscosity can be expressed with the Cox-Mertz law:

𝜼(𝜸.) = 𝜼∗(𝝎) 63. eq.

namely the dynamic viscosity is equal to the complex viscosity then the shear rate and the angle velocity is equivalent.

3. Principles of rheometry The aim of rheometry is the determination of the different rheological parameters. To decide that what is parameter required to examine depends on the material and the aim of evaluation. Based on the material the methods of rheometry can be classified into several groups taking into account numbers of considerations. As the main rheological types of materials are elastic, viscous and plastic, the general groups of methods are the viscometry of fluids (including viscous, plastic, viscoelastic and plastoelastic materials) and texture analysis of solid-line materials (including elastic, glassy, viscoelastic and plastoelastic materials). As it can be seen the classification is not strict; one kind of material can be tested by several methods. On the other hand, tests can be static and dynamic depending on the application of load (stress or deformation), the static test can be used for the evaluation of creep or recovery behaviour and the dynamic test can be used for the evaluation of storage and loss components of rheological properties, the determination of elasticity, viscosity or viscoelasticity of the material or the evaluation of the changes in the behaviour of material under a long-term load or changing temperature.

Maybe the simplest classification of methods by parameters and methods is the following:

• viscometry capillary viscometers falling ball viscometers rotational viscometers

• oscillatory testing • texture analysers, texture measurement

measurement of force measurement of distance measurement of volume measurement of time

The methods of rheometry can be classified into three groups based on the use. The fundamental rheological tests are aimed to determine the different rheological properties (for example dynamic viscosity of a material, the yield point of a plastic material or the Young’s modulus of an elastic material), but in practice not the exact values of parameters are important to know but empirical or technological values are more important to know. The empirical tests do not focus on all the rheological parameters only give practical information about the behaviour of material under specific conditions. For example, not the exact value of viscosity of a fluid is important to know but the ratio of viscosity change under specific conditions, e.g. by the change in concentration or the change in flow speed in different temperature. The third group is the group of imitative tests where the technological process or real effect is simulated and the behaviour of material is measured by practical aspects. Typical technological tests are the ones used in wheat flour analysis, for example Alveograph test which does not result basic rheological parameters but values easy to understand and apply in processing. All these methods have advantages and disadvantages; the fundamental tests give real rheological information but it is hard or more difficult to use in practice. In contrast, imitative tests do not have results about the fundamental base of behaviour but simple to perform and the result is easy to understand and apply in practice, therefore very useful in everyday quality control and monitoring the process.

3.1. Viscometry The viscometers are used to determine the viscosity of a fluid. Based on the design, the material can be in stationary conditions and the instrument makes it moving (e.g. rotational viscometers) or the material can be flow (e.g. capillary viscometers). In both case it is very important to control the pressure and temperature strictly because the viscosity may strongly change by it: for example the viscosity of water can be measured by 0.1% accuracy when the temperature is controlled with an accuracy of 0.04°C. On the other hand, several fundamental corrections are developed to calculate the viscosity values of specific materials based on data measured on specific temperature what makes possible to use these methods in industrial conditions.

3.1.1. Capillary viscometers In the case of capillary viscometers the material is moving in a stable positioned capillary tube. The advantages of these methods are: the conditions in the tube are well known because of the hydrodynamic bases; it is easy to control the external conditions due to the small volumes; rapid and results comparable readings.

The mathematical base of capillary viscometers is the Hagen-Poiseuille’s law:

𝑽 = 𝟏𝜼𝝅𝒓𝟒𝜟𝒑𝒕𝟖𝒍

64. eq.

what defines that the volume of a material which flows across in a tube in t time by the effect of Δp driving pressure difference is directly proportional to the radius of tube (r) and inversely proportional to the length (l) of capillary and the dynamic viscosity of material (η). If the viscometer tube is vertical the driving pressure difference is originated from the mass of material (therefore its volume and density) and the gravitation velocity; while the capillary has constant radius and length and only the time necessary for a specific volume to flow is measured. The shear rate is maximal in the axis of tube and minimal in its surface. This case it can be stated that the flowing is parallel to the axis (laminar flow) and axisymmetric one. Further requirements or presumptions are that the material is uncompressible, the shear rate is zero on the surface of capillary and it is depending on the actual stress in the different parts of the tube. On the other hand, while the material flows in the capillary, its mass (therefore the driving pressure difference) is continuously decrease what makes it very important that the ratio of volume of flowing material should be high in comparison to the volume of differential pressure forming volume. In practice the capillary viscometers have two or more reservoirs to provide the required volume. When the material flows from the starting reservoir to the capillary it is necessary to attenuate or eliminate the turbulent flow caused by the decrease in radius.

The simplest capillary type viscometer is the Ostwald viscometer (Figure 37.). It is also gives the fundamental base of the other viscometers. The reservoir of viscometer marked with B is filled with the material between the two upper dashed lines. For this, a rubbery stopper is placed in the right end of U tube and the material is pipetted across the U tube from the left side. When the B reservoir is filled, the viscometer is immersed into water bath to maintain constant and required temperature and when it is reached the suction force is terminated and the material starts to flow in the capillary part marked with C. The test ends when the meniscus of material reaches a predefined place. The result is the time what is required for the material flowing through mark to mark.

37. Figure: Ostwald type viscometer

The viscometers have their K factor or instrument conversion factor which is summarize the geometric properties of the instrument. By this, the dynamic viscosity can be calculated as

𝜼 = 𝑲𝝆𝒕 65. eq.

where ρ is the density of material. When the dynamic viscosity is not important or the density cannot be determined, the kinematic viscosity can be calculated as

𝝂 = 𝑲𝒕 66. eq.

Other widely used capillary viscometers are the Cannon-Fenske and the Ubbelohde viscometers. The Engler viscometer or the Sayboltviscometer are different from them as they are not U tube, but the flowing material leaves the instrument during evaluation (draining viscometers). During evaluation it is measured that what time is required to drain a specific volume of the material to a flask or measuring cylinder. These methods are generally indirect ones; relative viscosity values are determined and the reference material is generally water. The result of measurement by Engler viscometer can be calculated as:

°𝑬 = 𝒕𝒕𝒘

67. eq.

where t is the time of flow out of material and tw is the time of flow out of water. Knowing the rheological parameters of water the kinematic viscosity can be calculated as:

𝝂 = °𝑬𝟕,𝟔�𝟏−𝟏

°𝑬𝑩� 68. eq.

where B is a constant.

In practice several corrections are required to apply for gaining exact result. These are used to correct the kinetic energy loss of the bottom of U tube, the change in the volume of material maintain the driving pressure, the change of meniscus shape, the thermal energy develop from the flow of material, the turbulence occurs in the beginning and the end of capillary and the lack of

slippery at the wall of the capillary. On the other hand, capillary viscometers are relatively cheap ones, rapid, having minimal operational costs, easy to use and having good repeatability and these properties make them widely used.

For fluids with viscosity and suspensions such as fruit juices tube viscometers are also used. Their operating principle is similar to the capillary viscometers but they have got higher sizes and volumes; their length is 50 cm on average and the radius is between 3 and 15 mm and the horizontal arrangements are more frequent than the horizontal. In the case of tube viscometers the volumetric flow rate is measured.

3.1.2. Falling ball viscometers In the case of falling ball viscometers the analysed material is in stationary state in a tube and a body with known physical properties falling down in it. The most well-known falling ball type viscometer is the Höppler viscometer, where an iron or glass ball with a diameter between 11 and 15 mm is sinking in the liquid in a glass tube. The tube has a specific inclination what provides stable and repeatable measurement conditions. The measure parameter is the time what is required for the ball to fall down between two marks with 50 or 100 mm distance. The dynamic viscosity of the material can be calculated as:

𝜼 = 𝟐𝒈(𝝔𝒃−𝝔𝒍)𝒓𝟐𝟗𝝊

69. eq.

whereρb is the density of ball, ρl is the density of liquid, r is the radius if ball, v is the speed of ball. As the speed of ball is the quotient of l (the length of movement) and t (the time of falling down) the viscosity can be expressed as:

𝜼 = 𝟐𝒈(𝝔𝒃−𝝔𝒍)𝒓𝟐𝟗𝒍

𝒕 70. eq.

As the length, the gravitational acceleration and the radius of ball are known they can be expressed as a constant (K) and the viscosity is:

𝜼 = 𝒕(𝝔𝒃 − 𝝔𝒍)𝑲 71. eq.

Around the tube holding the analysed material thermostating liquid flows to provide constant and predefined temperature in the typical measurement temperature range (between -60 and 150°C, depending on the control liquid). As the density of material changes by the change of temperature, when the measurement is done on different temperatures different density values are required to determine or corrections have to be applied.

3.1.3. Rotational viscometers The principle of rotational viscometry is that the analysed material is located between two surfaces from which one is rotating and the second one is in stationary position or rotated by the viscosity of the analysed material. Basically the viscometer measures the torque on the surface in stationary phase. Based on the settings these applications it can be used for measurement in controlled stress (CS) tests when the stress causes rotational movement is constant and the shear rate is recorded or

controlled shear rate(CR)tests when the stress is measured what required to maintain a constant shear rate. The CS type rotational viscometers have better accuracies in the case of low shear rate and preferred to use in the case of strongly non-newtonian, plastic and elastic materials.

Based on the kind of surfaces one base types are

- coaxial cylinders (or cup and bob) - cone and plate - cone and cone - parallel plate (plate and plate)

In the case of coaxial cylinders typean inner surface of cylinder of a cup and an outer surface of cylinder of a bob do the measurement (38. Figure). Based on the construction of equipment twotypes can bedistinguished: in the case of Searle type the unrotated surface of cup is in fixed position and the bob rotates with controlled angular velocity. The rotational speed is added to the cup by the analyzed material resulting a torque against the rotation. The angular velocity and the torque are measured on the same axis. In the case of Couette type the cup is rotated by controlled angular speed what makes a torque on the bob and rotate it and the angular velocity is measured on the cup and the torque is on the bob (Figure 38.).

38. Figure: Coaxialrotationalviscometer (Searle and Couettetype)

The stress can be calculated as:

𝝈 = 𝑴𝟐𝒓𝟐𝚷

72. eq.

where M is the torque on the surcafe of a cyinder rotated by ω angular velocity and r is the radius of cylinder. The shear strain is:

𝜸 = (𝒓+𝚫𝒓)𝚫𝛚𝚫𝐭𝚫𝐫

73. eq.

whereΔr is the difference in radiuses of two cylinder surface, Δω is the difference in angular velocities and Δt is the delay in the rotation of the two cylinders. The shear rate is:

𝜸. = 𝒓 𝒅𝝎𝒅𝒓

74. eq.

and the viscosity is for newtonian materials is:

𝜼 = 𝑴𝟒𝚷𝝎

� 𝟏𝒓𝟏𝟐 −

𝟏𝒓𝟐𝟐� 75. eq.

where r1 is the radius of bob and r2 is the radius of cup. Different calculation methods are developed for the materials with different rheological behaviour. In order to simplify, the parameters of equipment are expressed as an instrument constant (K), for example the calculation for the viscosity of Newtonian liquids (69. eq.) is simplified as:

𝜼 = 𝑲𝑴

𝝎 76. eq.

39. Figure: Cone and plateviscometer

In the case of cone and plate viscometers (Figure 39.) a plate and a cone with small angle are the surfaces. One of them is rotated (generally the cone) and the other one is in stationary stage (the plate). The angle between cone and plate is small typically, it is between 0.5 and 5°. The viscosity can be expressed as:

𝜼 = 𝟑𝜶𝟐𝑹𝟑𝝅

𝑴𝝎

=K𝑴𝝎

77. eq.

where R is the radius of cone and plate. The advantage of this type of rotational viscometers is that the shear rate is uniform in all points and therefore the end-effects are avoided. In the case of parallel plates viscometer the material is placed between two disc-shapedplates and one of them is rotated with controlled angular velocity and the torque is measured on the second one. In the case of the cone-cone viscometers both surface is a cone.

The most frequent sources of errors in measurement by rotational viscometers are the slippery on the surface, the thermal loss caused by the internal friction and the end-effect. Several calculations were developed to minimize or eliminate the effects of these errors in the measurement.

In practice coaxial cylinders and cone and plate viscometers are the most frequently used ones. They advantages are the opportunity of use in wide operating conditions, the shear rate can be varied

steplessly from very low to very high values, a long-lasting continuous measurement can be performed applying changes in shear rate or temperature therefore the time-, temperature- and shear-dependent behaviour of materials also can be evaluated.

3.2. Oscillatory testing The principle of the oscillatory tests was presented in Chapter2.5. briefly, a periodically changing dynamic stress or deformation is applied on one side of the tested material and time-dependent deformation or stress is measured continuously on the opposite side. The phase angle shift between the stress and deformation give information about the kind of viscoelasticity and shear storage, shear loss and complex moduli and compliances can be calculated and the influencing factors also can be evaluated.

The oscillatory testing instruments can be classified into three main groups. The vibrational viscometers have got an oscillating measurement part which immerses into the analysed material, which is in stationary stage or continuously flows. The second type is the one when the stationer material flow is oscillated by excitation and the viscosity of material is measured. The third type is the simulation methods when the practical or industrial processes are modelled, therefore results practical reading but in several cases give no information about the basic rheological parameters. The most frequent oscillatory testing equipments have rotational constructions.

The generally evaluated parameters of oscillatory tests are G’, G” and G* moduli and η’, η” and η* viscosity values. The tests can be classified into the following groups:

- constant frequency and amplitude tests

Constant oscillatory stress or deformation is applied on the material and the moduli or viscosity values are determined. The frequency and amplitude is not changing during the test therefore the parameters characterizes type of viscoelasticity of the material in a relatively short term period (35. Figure and 36.Figure).

- time-dependent test with constant frequency and amplitude

When the frequency and the amplitude of dynamic load is constant but it lasts for a longer time it makes possible to analyse how the viscosity and moduli or compliance values changes by time. The general changes in behaviour can be seen in Figure 40. The material marked by 1 shows no change in moduli or viscosity under the ling-lasting load, therefore can be characterized as rheologically stable material. The material marked by 2 shows increase in shear moduli (therefore decrease in shear compliance), indicating that the constant dynamic load makes its structure more “harder”; its stability increases by time. The material marked by 3 shows decay or disintegration in structure indicated by the decrease in shear modulus value.

40. Figure: Change of time-stress and time-shear modulus duringconstantfrequency and amplitudetime-dependentoscillatory test

- temperature-dependent test with constant frequency and amplitude

When the frequency and the amplitude of dynamic load is constant but the temperature of material continuously changes it makes possible to analyse how the viscosity and moduli or compliance values changes by temperature. These tests also contain the time-dependent effects therefore these types of tests only supplement the time-dependent test and cannot be done in itself. These results give information that what viscosity and shear modulus values can be experienced in different temperature region and how homogeneous the structure of material is and how strong the bounds what stabilizes it.

- constant frequency and increasing amplitude tests

Tests with constant frequency and increasing amplitude make possible the rheological evaluation of materials under continuously increasing dynamic load. This means that the higher and higher amplitude result higher shear deformation and increasing shear rate. These tests are used to evaluate the range of elasticity or the linear viscoelastic range of material.

- increasing frequency and constant amplitude tests

Tests with increasing frequency but constant amplitude make possible the rheological evaluation of materials under continuously increasing shear rate. The constant amplitude causes the same shear deformation but the increasing frequency results increasing shear rate and as the shear rate increases the test makes possible to evaluate the changes in storage and loss moduli and the phase shift angle resulting data about the structure of material.

3.3. Texture analysers The texture analysers generally used to evaluate materials with high viscosity (viscous and viscoelastic) or to the evaluation of solid-like (plastic, plastoelastic and elastic) materials. The

2

1

3

fundaments of the test are similar to the ones which were presented in the case of viscosity. The Texture analysers are suitable to establish a constant deformation or constant deformation speed and a force measurement unit records the amount of force what is necessary to maintain or provide the deformation or what can be experienced when the deformation is applied on the material. Based on these considerations several types of use can be listed, but the most frequents are the followings:

- measurement of force: the material is loaded with a deformation (generally a constant deformation speed) and the force which is required to maintain the deformation. For example we cut a standard size part of a vegetable and the required force is determined.

- measurement of distance: the material is loaded with a constant stress (by the application of a force on a specific surface) or constant shear rate and the amount of deformation what the stress results is measured. An example for this type is the penetrometer used for the determination of maturity stage of apple.

- measurement of time: a specified stress or deformation is applied on the material and the time is measured from the application of probe to the final adaptation of material to the new circumstances (for example stress relaxation).

- measurement of energy: the evaluation of how much energy is required to maintain a status. For example how much energy is required for the final deformation for the tearing of an inflated dough disk (Alveograph test)

- ratio: when a stress or deformation on a material is applied more than one time sequentially what kind of change can be experienced in the response of the material.

The texture analysers have two main groups: universal and specific ones. The universal texture analysers can be used for several aims with different probes connected to a threaded socket (e.g. TA.XT texture analyzer). The specific equipment can be used for one specific purpose (or a few specific purposes) and generally not basic rheological parameters but practically useful results can be achieved (e.g. Farinograph). The universal analyzers can be used for specific purposes too usually thanks to the wide range of commercially available probes, but own methods and test probes are easy to made.

The texture measurement methods can be classified by the type of probe and the deformation it does. The texture analyzer measures force therefore the measurement is independent from the size and shape of test probe. Based on the probes and their application the following groups of tests can be differentiated:

- puncture: the texture analyser measures the force is required to push the probe into the material. The puncture can be applied until the probe immerses to a predefined distance from the surface or until the probe passes through (force measurement). On the other hand, the distance is also can be determined what is experienced when the probe loads the material by a predefined force (distance measurement). When the probe is selected or developed its area and perimeter are also important: the size of the area will determine the stress caused by the force (σ=F/A), but when the material is destructed the shear deformation will be experienced on the length of perimeter of probe. It follows that the same perimeter with different areas will not result the same readings. On the other hand, the deformation speed has to be constant. Puncture tests give relevant information on the elastic limit and the hardness of the materials.

- compression: similar to the puncture tests but in the case of compression tests the size (area) of test probe is larger the area of sample, therefore the whole deformed cross-section is under the one directional stress. The test is aimed to determine the compressibility of the material or the limit of fracture

- bulk compression: the pressure is applied on the whole surface of the material. The bulk compression tolerance is important to know when the material is exposed to the effect of hydrostatic pressure, for example HHP treatments or sterilizer.

- cutting or cutting-shear: the probe is used to cut a sample to two or more parts. The aim of these tests is the knowledge of amount of cutting force which is required when the material is sliced in one plane.

- shear: the shear deformation is often used for the determination of cutting force too, but in this case the force deforms the material not in one plane but pure shear occurs. In practice the cutting-shear is often found to be same as the pure shear.

- tensile: the material is stretched in one plane and the extensibility, stretchingdisposition, the stretching resistance, reversibility or fracture limit is determined.

- torsion: the material is exposed to a torque and the ratio of torsion under specific stress or deformation, the elasticity (reversibility)and fracture limit is the measured parameters.

The typical texture analysis curve can be seen on Figure 41. Its most informative parameters are:

• Hardness: the maximum force measured under the first deformation. On often case it characterizes the hardness of specimen.

• Fracturability: the first break in the texture analysis curve. It shows that the behaviour of material changes, a limit in rheological behaviour is achieved, for example the end of elastic (reversible) region.

• Adhesiveness: when the test probe moves upwards after deformation some materials performs a counterforce against the removal due to the stickiness or adhesion.

• Springiness: the ratio of the time of deformation of the second measurement and the time of deformation of the first measurement. It shows that how much ratio of original elasticity remained in the specimen after the first deformation.

• Cohesiveness: the ratio of the area under the curve in the second deformation and the area under the curve in the first deformation. It refers about the remaining elasticity of material under a repeating deformation.

• Chewiness: the sum of hardness and cohesiveness. It is a typical parameter for materials with high adhesiveness.

41. Figure: Typicaltextureanalysiscurve (compressionmode)

Based on the use almost every cases it has opportunity to do the measurement in force, distance or time mode, but the energy of deformation and the ratio also can be calculated. Beside the instrumental analyses several other experimental tests are available to test the materials. One of the simplest tests is the falling test when a product specimen is dropped from a specific height and the possible deformation effect is evaluated visually. Similar to this the sorting method used in food industry when the pieces of raw material is dropped to a sloping surface and based on the maturity stage and the texture the pieces snap to closer or further.

The advantages of texture measurements are the easiness, rapidity, applicability on heterogeneous materials, wide range of application fields, easy to develop new methods and probes and does not require high operational costs. On the other hand, the measured force has to be correlated to one or more relevant parameter, for example sensory experience, in order to gain usable results. The heterogeneous materials show high deviations in readings, therefore several repeats are suggested to done in order to improve the statistical relevance of data.

4. The use of rheology in food processing and analysis

The rheology is very important both for the production and the qualification of food products. First, the operated technical equipments have to be designed and selected by the consideration of the materials being under process. Different types of pumps, pistons, mixers or fillers, even pipelines are required for the materials with different rheological properties. For example, several references describe the modelling of flow of fluids in pipelines with different cross sections and areas. For example, when a material flows in a circular cross section pipeline rheologically it can be considered as a fluid under mechanical shear, where the shear force came from the flow velocity. When other influencing factors (such as gravity or the turbulence due to the not absolutely plain inner surface of pipeline) are not considered, the flow rate will be maximal in the central axis of the pipeline and it will be minimal on the surface. On the other hand, the shear rate is not zero on the surface but the material slips on it, therefore the friction of liquid molecules on the solid surface has to be also considered. The shear rate of oils flowing in pipeline is the following:

𝛾 . = 𝑄2𝑟𝜋

78. eq.

where Q is the flow rate [cm3/s] and r is the radius of pipeline (Tóth, 2000).He also bring the example of the sedimentation of suspensions: the shear rate can be calculated as:

𝛾 . = 𝑑2𝑔18𝜂

(𝜌0 − 𝜌1) 79. eq.

where g is the acceleration of gravity, η is the viscosity of liquid, ρ0 is the density of dispersed material and ρ0 is the density of liquid.

The rheological funds also can help to define the requirements on the type and ratio of raw material (e.g. the hardness of marmalade or the consistency of yoghurt) and to create check points both for in-process quality control and the testing of end-products (for example, in the first case an ingredients can be added when the viscosity of the intermediate product reaches a predefined value and in the second case the product is suitable when its shear yield is within a specific range). Rheology is also a took to meet the consumer’s sensory demands as several sense can be described by rheological parameters for specific products (for example the liquid is flowing instead of having a gel structure or a product is too soft during chewing or the crispiness of a foodstuff lasts no long time). It is also can be used in product development – helps to define the demands for engineers and developers and it also makes possible to compare different unit operations, technological variations, raw material substitutions or give a new property for a product.

4.1. Rheological methods in cereal analysis There are several specific rheological methods for the evaluation of cereal and cereal products, especially wheat flour and dough. The gluten and starch are the two components which determine the rheological behaviour of these products. The gluten forms a more or less strong three dimensional network in the presence of water with the absorption of large amounts of water. The formed gluten network gives extensibility and strength to the dough which basically determines the

quality of both raw materials and products. On the other hand, when the dough is heated the starch starts to gelatinize which result rapid increase in the viscosity of flour-water system.

The most frequently used rheological tests in flour analysis are the Alveograph, Farinograph and Estensigraph tests and the Falling number analysis, but several other methods and equipment are used (for example Mixograph, Rheofermentometer, Amylograph, rheometer, etc.). The Alveograph and Extensigraph are used to evaluate the resistance of properly prepared dough against extension, therefore they can be characterized as tensile tests. The Farinograph and Mixograph was found to be torsion tests as the dough is kneaded during the evaluation and both pressure and shear deforms it. The Falling number test basically based on the principles of falling ball viscometry and the Amylograph test is a rotational viscometric method with controlled shear rate.

The Alveograph test is a biaxial stretching using constant 2 to 1 ratio of flour and 2.5% NaCl solution, taking into account the moisture content of flour. The dough is mixed, formed, rested then tied down a clamping device and inflated until it gets torn. The equipment draws a curve with the pressure in the dough bubble on the y axis and time on the x axis (Figure 42.). The height of the curve is the P value (expressed in mm) characterizes the strength of the dough; this is the maximum pressure to what the dough can resist; the length of the curve is the time from the starting of deformation until the dough gets torn – this value expressed in mm is the L value characterizes the extensibility of dough. Their ratio is the P/L value, the formal quotient of curve and the area under the curve is the W value expressed in 10-4J, the work required for the final deformation of dough. The international recommendations contain reference values for different uses of flour.

42. Figure: RepresentativeAlveograph diagram

The Farinograph test evaluates the dough in dynamic conditions by continuous kneading using two z-arm mixer. The equipment records the resistance of dough against the deformation and displays a diagram (Figure 43.). The curve has upper and lower limit lines due to the attenuation of the arms and a drawn midline what averages the values presented by the upper and lower line. The test does not apply the same ratios of flour and water, but requires a specific resistance to adjust with the addition of water to the same amount of flour. This specific resistance is 500 BU (Brabender Unit) what is an arbitrary unit of viscosity. The more water addition results weaker the less water addition results stronger dough. The amount of water expressed in percentage is the water absorption

P

extensibility (mm)

areaunderthecurve

height of curve

L

capacity, what is the amount of water used in bread making. Its value is ranged from 55 to 70%, the higher values are more advantageous. It is hard to add as much water as necessary to reach the 500 BU exactly therefore a small tolerance is allowed; the obtained maximum resistance line has to be within 20 BU from the 500 BU line. In the international evaluation the readings are the arrival time (the time when the upper line of curve crosses the maximum resistance line), the peak time (the time when the curve reaches its maximum value), the mixing tolerance index (the difference of resistace values measured in the peak time and 4 minutes after), the departure time (the time when upper line goes above the maximum resistance line), stability (the time between the departure and arrival times) and Farinograph Quality Number (the tenfold of the time expressed in seconds from the beginning of evaluation to the time when the upper line falls 30 BU below the maximum consistency line). In Hungary, the dough development time (the time when the midline reaches its maximum value), stability (the time while the midline and the maximum resistance line are parallel to each other), degree of softening (the distance between the midline and the maximum tolerance line in BU) and the baking value (calculated from the area between the maximum resistance line and the midline) are the evaluated parameters

43. Figure: RepresentativeFarinograph diagram

The Extensigraph is an uniaxial extension. A dough with 500 BU consistency is prepared using flour and 2% NaCl solution in the bowl of the Farinograph, formed to a cylinder, rested for 45 minutes, then its ends are impacted and its middle part is pulled off with a constant speed until the dough breaks. Next, the dough is formed again and the resting and extension is repeated two times (in the 90 and 135 minutes). The three deformations are recorded on three diagrams (Figure 44), there the x axis is the time starting from the deformation, characterizes the extensibility of dough and the y axis is the resistance of dough against the extension. The quality parameters are the maximum resistance (the highest resistance value of the curve), the resistance to extension5cm (the resistance of dough

0 5 10 15 20

developmenttime degree of softening

stability

Brab

ende

r Uni

t

areabetweenthe maximum resistance line and themidline

of curve

measured in the 5th cm of the x axis), the extensibility (the length of the curve) and the deformation work (the area under the curve).

44. Figure: RepresentativeExtensigraph diagram

The Hagberg’s falling number method is used for the determination of amylase enzyme activity. It is an important quality issue because the high enzymatic activity makes good dough structure impossible due to the rapid increase in the fermentation activity of yeasts. During the evaluation a flour-water suspension in hot water bath is mixed for a minute, then the mixing rod falls in the starch suspension. This falling will be slow due to the fact that boiling water gelatinizes the starch in the suspension, but the amylases start to breakdown it. The higher amylase activity result faster starch degradation and faster decrease of viscosity, therefore the rod will be got down sooner. The falling number is the time in seconds from the beginning of mixing to the arrival of rod to the bottom of tube.

The Amylograph test is a complex evaluation of the starch properties of dough. The amylograph is a Searle-type rotational viscometer, actually. During the test a thin suspension is made from flour and destilled water with mixer and it is placed into a bowl what is the rotated part of the viscometer. The stationary part has eight pins immersing into the suspension. The bowl is heated during the examination; from a 25°C starting temperature it is raised by 1.5°C in every minute. The increasing temperature influences the properties of starch: when the temperature reaches the limit temperature of gelatinization the viscosity of suspension raises rapidly and tends to a maximum value. The further increase in temperature and the increasing enzymatic (amylolytic) activity result decrease in viscosity later and when the starch is degraded fully the viscosity value falls back to the minimum value. This method is used to the analysis of rye flours especially. The results of the test are the starting temperature of gelatinization, what gives important information about the thermal tolerance of dough, and the maximum temperature and viscosity of gelatinization (Figure 45.).

500 maximumresistanc

e

extensibility Resis

tanc

etoe

xten

sion

(BU

)

resistance to extension5 cm

0 5 10 15 20

45. Figure: RepresentativeAmylograph diagram

The bread crumb as final product can be evaluated by texture analyser and the result characterize the firmness and this value shows good correlation to the sensory experience, therefore it can be used to the instrumental analysis of sensory effect both during storage, modification of ingredients or the technological parameters. The test requires a cylinder probe with 38.1 mm diameter by the requirements of AACC 74-09 method. The texture analyser has to be set to force measurement. The pre-test speed whit what the probe moves to the bread slice until the touch is 1 mm/s and the deformation rate is 1.7 mm/s. The bread has to be sliced to 25 mm width slices and the probe moves to the depth of 25% of the slice. The test has to be performed far enough from the crust. The measured values are the crumb firmness what is the maximum force of the compression test, the stiffness what is the slope of the curve in the linear part and the relative elasticity what is quotient of the stored (remaining) force after twenty seconds of reaching the maximum firmness value and the maximum force. The deformation work can be also calculated by the integration of area under the curve from the starting point to the maximum firmness value (Figure 46.).

46. Figure: Representativetextureanalysis diagram of bread firmness

4.2. Rheological methods in fruit and vegetable analysis The hardness of skin or the whole fruit is a very important sensory parameter of fresh fruits and vegetables as it refers to the maturing stage, the suitability for entering a specific processing line in several cases and basic selection choice for the consumers. The hardness is a parameter what can be measured easily by texture analysis and the values show good correlation to the sensory experiences. The analysis of hardness can be performed by puncture or compression tests; the first one characterizes the skin hardness well, while the latter one is more useful in the evaluation of flesh firmness.

During the skin hardness test the texture analyser measures in compression mode. The specimen is placed on a flat platform ensuring the stable position and hindering the movement perpendicularly to the direction of deformation movement. The test probe is a thin needle (with a diameter of 2 mm generally). When the needle touches the surface of the fruit and start to move into the fruit it is done without destructing it, without puncturing or crushing the cells and tissues. This part of texture analysis curve is linear and shows the elastic behaviour of tissues. Later the force reaches the puncture limit and the test probe penetrates into the skin resulting change in the way of curve. This point is the bioyield point indicating the resistance of fruits against the mechanical load giving useful information about its load capacity, the transport tolerance or demands on handling. The way of curve generally shows a significant decrease in force after the puncture indicating that the flesh is more less tolerant against deformation than the skin (it is more soft, such as in the case of peach), but sometimes only a small decrease can be experienced (for example in the case the apples harvested for long term storage). The fruits with very soft flesh show low and stable or slightly increasing force values against the further deformation, but the force increases during the further analysis of products with hard flesh, especially due to the increasing friction between the cylinder surface of needle and the tissues of flesh.

The flesh hardness is also evaluated by puncture test, but the diameter of probe is higher (generally higher than 10 mm) and its surface is not pointed but has flat surface. It results that the test probe will deform not the skin primarily but the whole tissue layers under the affected area. The curve of flesh hardness test result similar curve; the initial part is a linear region where the deformation in reversible and the material is found to be elastic. Later a small sudden decrease is experienced when the intracellular parts of cells (e.g. membranes) and the cell walls are damaged and the tissue contents start to mix. This point is called bioyield. The further increase in force is caused by the further pressure of dehquescent fruit flesh pulp and it increase until the point when the fruit skin damages, cracks. The further increase is caused by the compression effect of further pressure on the flesh and as it cannot flow away it results continuous increase in force.

Another rheological issue regarding fruits and vegetables is the measurement of cutting force. It can be used for the determination of freshness, the evaluation of chewiness, but it also can be used for the comparison of effects of different technological parameters of the process (e.g. blanching). In this case cutting shear deformation is applied on the specimen while the other side is stablysituated in aflatsheet, or the force required to cut the specimen asunder can be also measured. The force-time graph is similar to the one of flesh hardness test; a linear region when the force is proportional

linearly to the deformation is followed by the bioyield point but the rapture point follows it suddenly. The counterforce against the following deformation is generally uniform when the thickness of specimen is constant

The measurement of viscosity is used for the analysis of juices, concentrates, pulps and jams too.Based on the wide range of possible analytical aims, the parameters of used methods can be changed in wide scale. Generally universal rotational viscometers are used what are easily suitable to measure of the value and changes of viscosity or apparent viscosity under constant external parameters or while the temperature or shear rate changes. The analysis under constant parameters makes the measurement of single parameters or the time-dependent behaviour possible. The continuously changing temperature enables the evaluation of temperature dependent changes, while the increase of shear rate enables to examine the limits and parts of viscoelastic behaviour. These measurements make possible to predict that how the material will change under the storage, under the effect of heat treatment or how the different additives can modify the behaviour.

4.3. Rheological methods in the analysis of meat and meat products The measurement of cutting shear force is the basic and maybe the earliest rheological method for the analysis of meat. Its test probe is a blade with 1.016 mm thickness and a V shape cutting surface with 60°opening angle. On the other side of specimen an open surface platform is placed what makes possible the total cutting of specimen. The meat samples have to be prepared properly providing constant diameter and appropriate length for the samples. For this purpose cylinder shaped specimens with 1.27 mm diameter or, in special cases, other diameters or prismatic or rectangular shaped specimens have to be made perpendicularly to the direction of meat fibres. The deformation speed is constant and the highest force against the cutting shear or the area under the force-time curve representation the cutting work can be the result of the test. Another test probe for the determination of meat texture is the Slice Shear Force test.This test uses a thin steel sheet with flat cutting edge and it basically developed for the evaluation of beef tenderness after cooking.The 5 or 10-bladed Kramer Shear Cell is also recommended to the rheological analysis of meat what performs shear strain on the specimen in 5 or 10 deformation planes. These tests simply and easily characterize the resistance of meat against chewing and can be used for the comparison of a wide range of influencing factors (variety of animal, breeding technology, animal nutrition, post mortem processes in meat, kitchen technology, additives and so on).

4.4. Rheological methods in the analysis of milk and milk products The viscosity of milk is influenced by the composition and concentrations of its solid and colloidal components and has strong correlation to the sensory value. From the chemical components the fat globules and protein micelles have significant positive effect on the viscosity and the latter one results the ageing of milk resulting increase in apparent viscosity.

Based on the literature the capillary, galling ball and rotational viscometers can all be useful in the texture analysis.

The products with thick liquid or gel structure such as yoghurt, mayonnaise or creams can be evaluated by back extrusion test. The test probe has two parts: a rig and a disc. The specimen is placed in the rig and it is positioned stable and concentrically to the axis of disk. During the test the disk is immersing and compressing the sample in the rig, resulting a simple compression first, but as the deformation and counterforce increases the specimen extruded out from the rig around the disk. By the proper selection of the diameters of rig and disk the thickness of extrusion can be set up to make the compression force proportional to the density of material determined by sensory tests. The highest force value in the compression stage refers to the firmness of specimen, and the force can be measured in the extrusion is correlate with its consistency (thinner or thicker character). When the disk is removed from the rig a negative counterforce is experienced what characterize the cohesiveness of the sample.

In the case of cheese the texture analysis aimed to determine the hardness of product. These tests are similar to the crumb hardness test of bread; the texture analyser works in compression mode and a spherical or cylinder probe is used to penetrate into the specimen. The highest force value of a constant deformation and deformation speed refers to the firmness of cheese. In the case of soft cheeses or cheese spreads a negative force can be experienced when the test probe is removed from the sample. It is caused by the adhesive character of the cheese. The shear force and cutting shear test probes of texture analysers are also used for the determination of firmness of cheese. For this purpose wire cutter probe or thin steel plan is often used. Oscillatory testing is also used in the analysis of cheese products for the analysis of linear viscoelastic region, the storage, loss and complex moduli and the limit of plasticity.

References AACCI Method 22-10.01: Measurement of alpha-Amylase Activity with the Amylograph

AACCI Method 54-10.01:Extensigraph Method, General

AACCI Method 54-22.01: Rheological Behavior of Flour by Farinograph

AACCI Method 54-30.02:Alveograph Method for Soft and Hard Wheat Flour

AACCI Method 74-09.01: Measurement of Bread Firmness by Universal Testing Machine

AACCI Method 74-10.02: Measurement of Bread Firmness -- Compression Test

AbangZaidel, D.N. Chin, N.L. ,Yusof, Y.A. (2010): A Review on Rheological Properties and Measurements of Dough and Gluten. Journal of Applied Sciences, 10. 20. 2478-2490.

Adebowale, A.A.: Food Rheology FST-310. http://unaab.edu.ng/attachments/455_FST%20310%20%20lecture%20note-DR%20ADEBOWALE.pdf

Application guide of TA.XT2.Stable Micro Systems, Ltd.. Hamilton

ARES-G2 Rheometer. TA instruments, www.tainstruments.com

Barnes H.A. (1997): Thixotropy – A review. Journal of Non-Newtonian Fluid Mechanics, 70.1-33.

Barnes, H.A. (2000): A Handbook of Elementary Rheology. University of Wales, Institute of Non-Newtonian Fluid Mechanics, 200.p.

Barnes, H.A., Hutton, J.F., Walters, K. (1993): A introductiontorheology. Elsevier Science Publishers, Amsterdam, 191. p.

Bourne. M.C. (2002): Food Texture and Viscosity: Concept and Measurement. Second Edition. Academic Press, London, 427. p.

Chin, N.L., Chan, S.M., Yusof, Y.A., Chuah, T.G., Talib, R.A. (2009): Modelling of rheological behaviour of pummelo juice concentrates using master-curve. Journal of Food Engineering, 93. 2. 134-140

Dak, M., Verma, R.C, Jaaffrey, S.N.A. (2007): Effect of temperature and concentration on rheological properties of ‘‘Kesar” mango juice. Journal of Food Engineering, 80. 4. 1011-1015.

Élelmiszerek, zöldségek, gyümölcsökreológiája . In: Laboratóriumimérések. ftp://physics2.kee.hu/_Fizika1/Laborgyakorlatok_leirasa.pdf

Foegeding, E.A., Vardhanabuti, B., Yang, X. (2011): Diary systems. In: Practical food rheology – An interpretive approach. Eds: Norton, I.T., Spyropoulos, F., Cox, P., Blackwell Publishing Ltd., Chincester, 133-172.

http://epa.oszk.hu/00000/00025/00003/gyumolcs.html

Introduction to polymers: 5.3.1 Time-temperature superposition. The Open University.http://www.open.edu/openlearn/science-maths-technology/science/chemistry/introduction-polymers/content-section-5.3.1

Lopes, A.S., Mattietto, R.A., Menezes, H.C., Silva, L.H.M., R.S.Pena (2013): Rheological behavior of Brazilian Cherry (Eugenia unifloraL.) pulp at pasteurization temperatures.Food Science and Technology (Campinas), 33. 1. 26-31.

Macsihin, J. A., Macsihin, Sz. A. (1987): Élelmiszeriparitermékekreológiája. MezőgazdaságiKiadó, Budapest, 248. p.

Ngai K.L., Capaccioli S., Plazek D.J. (2013): The viscoelastic behaviour of rubber and dynamics of blends. In: The Science and Technology of Rubber Eds.: Mark J.E., Erman B., Roland M. Elsevier Inc., Waltham, USA, 193-283.

Ross, D., Keeping, C. (2008) Measuring the Eating Quality of Meat. Food Marketing and Technology, 2008. February, 30-32.

Schramm, G. (1998):A practical approach to rheology and rheometry. Second edition, Gebrueder HAAKE Gmbh, Karlsruhe, 291.p.

The textureanalysisapplicationsdirectory.Stable Micro Systems, Ltd.. Godalming, 24. p.

Tóth S. (2000): Reológia, reometria. VeszprémiEgyetemiKiadó, Veszprém, 232. p.

Weipert, D. (2006):Fundamentals of Rheology and Spectrometry. In: Future of Flour: A Compendium of FlourImprovement, Ed.: Popper, L., Schäfer, W., Freund, W., ERLING Verlag GmbH & Co. KG, Clenze, 117-146.

Young, N.W.G. (2011): Introduction - Why the interpretive approach? In: Practical food rheology – An interpretive approach. Eds: Norton, I.T., Spyropoulos, F., Cox, P., Blackwell Publishing Ltd., Chincester, 1-28.