# Controlled-Stress Rheometry

- Introduction to Rheology
- Important Aspects of Rheometry
- Principles of Rheology
- Rheological definitions
- Ideal Elastic Solid Material
- Ideal Newtonian Liquid Material
- Ideal Plastic Material
- Viscoelastic Materials
- Dynamic Method
- Instrument Description
- Geometries
- Variables of interest
- Calibration
- Experimental Procedure
- Creating the Test Program
- Sample Preparation
- Loading the Sample
- Two Case Studies
- Fully Hydrogenated Soybean Oil
- Tristearin-Triolein Blends
- Rheological-Fractal Model
- Remarks
- References

## Introduction to Rheology

The name *rheology* refers to the *study* of flow and deformation of material when a certain force is applied. Rheology is applicable to all materials, from gases to solids. *Rheometry* is the term used for the *experimental characterization* of a material's rheological behaviour, although many people uses rheology as a synonymous. Theoretical aspects of rheology are concerned with the relationship between the stress (force) applied and the strain (deformation) or strain rate (flow) to the internal structure of the material. To learn about the internal structure using rheometry, a mathematical model is needed. Rheometry has been divided into two categories: large deformation and small deformation.

Experiments carry out in rheometers require that the material is in intimate contact with two surfaces that the instrument provides. One of this surfaces is fixed while the other moves. This way of measuring creates a gradient of velocities perpendicular to the two surfaces. The material in contact with the fixed instrumental surface is almost immobile while the material in contact with the other instrumental surface is always on the move. Hence, many of the experiment on a rheometers involved shear forces.

This section focus on small deformation obtained when a small-enough shear stress is applied. The section finishes with a brief explanation of the rheological-fractal model that relates the storage modulus *G'* with the amount of solid and the rheological fractal dimension.

## Important Aspects of Rheometry

The main important aspect when performing a rheometrical measurement are introduced in the following summary table

## Principles of Rheology

Edible fats are semi-solid materials made by liquid oils and solid fats. These systems have been shown to form a crystalline network that entraps liquid oil (Marangoni and Wesdorp, 2013), which allows them to behave like solids until a deforming stress exceeds a yield value. Once the deformation is larger than the yield value, the material starts to flow like a viscous liquid (de Man and Beers, 1987). Some of the textural properties in butter, margarine and spreads, like spreadability and hardness, are largely rheological in nature (Prentice, 1984).

Very hard fats, i.e. at very high SFC contents (80-100%) or the ones with a high melting point, display ideal solid Hookean behavior below their yield stress, while oils display mostly ideal Newtonian flow behavior. At intermediate SFC values (15-70%), fats display viscoelastic behavior, i.e., both elastic (solid) and viscous(liquid) properties.

Small deformation rheometers are designed to carry out transient and dynamic experiments. An example of a transient experiment is a creep one, where the applied force is constant and the strain is measured as a function of time. In dynamic experiments, an oscillatory stress (or strain) is applied and the corresponding oscillatory strain (or stress) response is measured. For example, each oscillation can be carried out with a slight increase or decrease of its amplitude or, if the amplitude is kept constant, then the frequency can be changed.

Dynamic methods are used to learn about the viscoelastic properties of the material. Two parameters to measure are the amplitude and phase angle difference δ between the input stress and the resulting strain waves. From those values, elastic moduli are obtained. The amplitude of the applied stress is such that correspond to *the region of interest*. The applied stress is usually very small, hence these experiments are named SAOS or small amplitude oscillatory shear. The most popular region of interest is the one where the ratio between the stress and the strain is constant. This region is named LVR, linear viscoelastic region. In the last couple of years, a new region of interest has emerged. This new region starts where the LVR finishes and as such it comprises non-linear responses of the material. To study this region, large amplitude oscillatory shear or LAOS needs to be used which is beyond the scope of this section.

## Rheological definitions

**Ideal Elastic Solid Material**

A material that exhibits ideal elastic behavior is referred to as a *Hookean *solid. An ideal solid is one that stores all of the energy when it is deformed. This energy is released upon removal of the applied stress. The proportionality constant between the deformation and the force is given by the relation in Eq. 1

**(Eq. 1)**

The elastic modulus is different according to the kind of stress applied. A simple compression or elongation uses the young modulus as the constant of proportionality. While a shear stress, σ uses the shear modulus *G* as the proportionality constant, as indicated in Eq. 2

*=*𝛾

**(Eq. 2)**

where 𝛾 is the shear strain force or deformation per unit area. Only elastic solids display Hookean behaviour. Some solid materials in foods behave as ideal solids only for small deformations. When Hookean behaviour is no longer observed, the stress is no longer proportional to the strain.

**Ideal Newtonian Liquid Material**

Liquids are isotropic so that, upon the application of a stress, the material moves without modification of the structure for as long as the stress remains on it. The characteristic property, in this case, is that the rate at which the material deforms is proportional to the applied stress.

The relation that holds for ideal Newtonian liquids is that the applied stress, σ, is proportional, via the viscosity *η*, to the strain rate *Ẏ* as seen in Eq. 3

*η*Ẏ

**(Eq. 3)**

The viscosity of non-Newtonian liquids is not constant and can depend on the rate and/or the time over which the shear force is applied as well as the temperature of the liquid.

**Ideal Plastic Material**

The rheological behaviour of an “ideal plastic” (or Bingham plastic) material is a function of a critical yield stress, σ_{c}. For values of stress smaller than σ_{c}, the application of a deforming force leads to elastic or solid-like behaviour of the sample. For values of stress greater than σ_{c}, the bonds in the sample start to break, leading to its plastic deformation or flow, described as a liquid behaviour of the sample.

The rheological behaviour of this ideal plastic is described by two Eq. 4

*γ*G (for < σ

_{c})

σ - σ

_{o }= Ẏ

*η*(for < σ

_{c})

**(Eq. 4)**

where ideal plastics show a clear division between elastic and viscous behaviour, and non-ideal plastics exhibit these two behaviours simultaneously.

**Viscoelastic Materials**

When a material behaves as elastic and viscous simultaneously, they behave as viscoelastic materials (Prentice, 1984). Edible fats exhibit viscoelastic behaviour that results from the presence of liquid fractions in the solid matrix (Marangoni and Wesdorp, 2013). The crystal network in an edible fat is formed at the smallest length scale by polycrystalline crystalline nanoparticles (CNP) (Acevedo and Marangoni 2010a). The CNPs aggregate to form larger fractal flocs in the nano to micro length scale. These aggregates form either separated larger crystals or they form larger clusters which eventually form the 3D solid network.

A deformation of the 3D crystal network will stretch the bonds between the flocs or aggregates. If the strain is within the elastic region, the material will respond as a solid, and thus the stretched bonds will return to their equilibrium positions. On the other hand, if the stress exceeds the elastic limit, bonds will break and the material will flow. It is well known that edible fats behave as fluids at strains beyond their limit of elasticity. Having in mind that broken bonds can reform once the stress is removed, the material will display viscoelastic behaviour (Marangoni and Wesdorp, 2013).

**Dynamic Method**

It is the dynamic method that enables the study of the viscoelastic behaviour of fats. As mentioned above, SAOS studies on fats are carried out in the LVR region.

Dynamic methods consist in applying a shear stress in an oscillatory way. Different situations can be applied: a fix frequency or many different ones, a fix stress amplitude, or a stress that increases the amplitude , one cycle or many repetitions. The selection of the right combination will depend on what the researchers are trying to study. The basic theory is introduced here.

When a sinusoidal perturbation is applied to the sample, the system responds with an oscillatory strain, Eq. 5:

*γ*=

*γ*

_{o }sin(

*ω t*)

**(Eq. 5)**

- where γo is the maximum amplitude of the strain, *ω* is the oscillation frequency expressed in rad/s (in units of hertz, it is *ω/2π* hertz) and *t* represents time. If non-linear processes are ignored, so that the system is represented by a differential equation involving a simple harmonic oscillator (energy storage term) and a friction (energy lost) term that is linear in velocity, then the oscillatory strain produces a shear stress, σ, acting on a surface in contact with the material. This has been described by Marangoni and Wesdorp (Chapter 6, 2013), and is given by Eq. 6

_{o }sin(

*ω t + 𝛿*)

**(Eq. 6)**

Here *σ*_{o} is the maximum amplitude of the shear stress and *𝛿* is the phase angle shift of the stress with respect to the strain. Fig. 1 shows the curve of an oscillating stress and the resulting oscillating strain as functions of time.

For an ideal (or purely elastic) solid, the maximum strain occurs when the maximum stress is applied. The stress and strain are said to be in phase when δ = 0° (Fig. 1). If the material is purely viscous, the stress and strain are out-of-phase by 90° (Fig. 1). Viscoelastic materials exhibit a behaviour that lies somewhere in between these two extremes, where the phase shift is between 0° and 90° (Fig. 1) (Steffe, 1996).

Eq. 6 can be manipulated using trigonometric identities to obtain:

_{o }sin(

*ω t*) cos(𝛿)

*+*σ

_{o }sin

*(*cos(

*𝛿*)*ω t*)

**(Eq.7)**

- which can also be rewritten as:

*Y*

_{o}[σ

_{o}/y

_{o}cos(𝛿)] sin(

*ω t*) + [σ

_{o}/y

_{o }sin(𝛿)] cos(

*ω t*)

**(Eg.8)**

The above equations allow a definition of dynamic moduli as:

*G*' =

*σ*

_{o}/*y*cos(δ)

_{o}**(Eg.9)**

*G*" = σ

_{o}/

*γ*

_{o}sin(

*δ*)

**(Eg.10)**

where *G'* is the shear storage modulus, which provides information about the elastic nature of the material, and *G"* is the shear loss modulus, which provides information about the dissipation of energy by the material.

Eq. 8 can then be written to obtain:

_{o }G' sin(

*ω t*) +

*" cos(*

*γ*_{o }G*ω t*)

**(Eq. 11)**

*G'* and *G"* are the variables measured in fats experiments. A purely elastic material has *G"* = 0 for which δ = 0°, while a purely viscous material has *G"* = 0 for which δ = 90°.

The complex modulus, *G* *combines both *G' *and *G"* contributions and is expressed as a complex number, *G** = *G'* + *ίG*" The norm, or absolute value, of a complex number is calculated as the square root of the sum of the squares of the real and imaginary parts of the complex number, namely *G'* and *G"* . Thus, the absolute value of the complex modulus is given by √*G' ^{2} *+

*G"*which corresponds to the maximum stress over the maximum strain, namely:

^{2}*G'*+

^{2}*G"*√(σ

^{2}_{0}/γ

_{o}cos 𝛿)

^{2}+(σ

_{0}/γ

_{o}sin 𝛿)

^{2}

**(Eq. 12)**

The ratio of these two moduli corresponds to the tangent of the phase angle and is a useful measure to characterize the viscoelastic behaviour of a particular sample. It is given by:

*(*𝛿

*)*=

*G*"/

*G*'

**(Eq. 13)**

tan*(*𝛿*)* is directly related to the energy lost divided by the energy stored.

Many food products containing fats, like chocolates, spreads or even gels, can be studied using the dynamic methods that allow the characterization of *G'*, *G*" and tan*(*𝛿*)* (Whorlow, 1980). Fig. 2 shows the behaviour of *G'* and *G*" for three different cases: (A) a stabilized dispersion, (B) a weakly flocculated dispersion, (C) a strongly flocculated dispersion or gel.

The first case, Fig. 2A shows a stabilized dispersion, the simplest case where interparticle forces are negligible between the black dots (particles) immersed in the white medium in such a way that the system is stable. A log-log plot of the values for *G'* and *G"* as function of the angular frequency shows two straight lines, or slopes. Slope values can be used to compare which of the two moduli (*G'* or *G*") will dominate at high or low frequencies. The second case in Fig. 2B shows the case where flocs (black and grey dots sticking together) are present. The interparticle forces between basic particles are now stronger compared to the first case. A log-log plot of *G' *and *G"* as functions of frequency helps to understand what is happening. *G' > G"* at high frequencies but remains smaller for low frequencies. The behaviours of both and show the viscoelastic nature of the material. The third case in Fig. 2C is for very strong interparticle forces that creates a 3D network. The results show a frequency-independent elastic modulus

*G'*that largely exceeds the loss modulus

*G"*. It can be seen that at high frequencies the value of

*G"*increases. Fig. 2C does not show it, but when the yield point is reached,

*G'*decreases abruptly and increases, so that the two moduli cross, marking the breaking point of the material.

## Instrument Description

Rheometers can be of two types: controlled-stress and controlled-strain. A controlled-stress rheometer applies a torque to either control the stress at a desired level or to drive the strain to a desired amount. In this kind of rheometer the material is placed between two surfaces: a bottom surface, which is fixed, and a top surface, called the geometry, which rotates. In the controlled-stress rheometer, the torque or stress is the independent variable and is applied to the geometry. In the controlled-strain rate rheometer, the material is placed between two plates. The bottom plate moves at a fixed speed and the torsional force produced on the top plate is measured. Hence, for this case, the strain rate is the independent variable and the stress the dependent one. Here it is worth noting that a controlled-stress rheometer is a better approach for determining yield stress. With these instruments, the stress can be increased in a gradual, controlled way until the yield point is reached. With controlled-strain instruments, the yield point has to be exceeded before the corresponding stress can be determined (Semanicik). Fig. 3 shows a schematic representation of the geometry and the moving parts in a controlled-stress instrument.

The measurement of the angular displacement is carried out by using an optical encoder device, which can detect very small movement, as small as 40 nRad. The encoder consists of a non-contacting light source and photocell arranged on either side of a transparent disc attached to the drive shaft. There is also a stationary segment of a similar disc between the light source and encoder disc. The interaction of these two discs results in light patterns that are detected by the photocell. As the encoder disc moves when the sample is strained by the applied stress, these patterns change (AR2000 Rheometer, 2005). The associated circuitry interpolates and digitizes the resulting signal to produce digital data. This data is directly related to the angular deflection of the disc, and therefore the strain of the sample.

Fig. 4 shows a TA AR2000 (TA instruments, New Castle Deleware, USA) rotational controlled-stress rheometer. This has three main components: (a) the main unit mounted on a cast metal stand that supports the geometry, (b) an electronic control circuitry contained in a separated box (electronic box) and (c) the sample holder (Peltier plate).

The main unit uses an arm that can be moved up and down, either directly with the buttons on the front panel of the main unit or using the software. This arm contains the optical encoder and motor shown in Fig. 3. The electronic box is the link between the rheometer and the computer. The sample holder consists of a Peltier plate that uses the Peltier principle to control the temperature. A water bath helps in cooling this device. The Peltier plate can be removed and replaced by concentric cylinders (Fig. 5). Concentric cylinders are used with fluid materials for which the temperature must be controlled since the system consists of a water jacket and an inner cup where the sample is placed. The geometry then slides into this cup.

During testing, the sample is in contact with two surfaces: a static bottom plate and a moving geometry. Since the bottom plate does not rotate, the key to a useful measurement is the selection of the geometry.

**Geometries**

The geometry is the piece that is attached to the driving motor spindle and rotates under specific conditions. Geometries are constructed from stainless steel, aluminum or acrylic. Stainless steel is relatively heavy but has a low coefficient of thermal expansion. Aluminum is less useful because it has a higher thermal coefficient of expansion and is chemically incompatible with many substances. Acrylic and polycarbonate material offer less resistance but can react chemically with the sample under study. Typical geometries are parallel plate, cone and plate and concentric cylinders (Fig. 5).

Concentric cylinders are usually used for low-viscosity liquids; cone-and-plate are used for liquids and dispersions with particles size less than 5 µm; and parallel plates are used for gels, pastes, soft solids and polymer melts. The rheometer uses the geometry specifications (i.e. diameter, angle of cone, gap) to give a result that is independent of the geometry used. Geometries come in different diameters: larger diameter geometries (60 mm, 40 mm) are used for low viscosity materials, while smaller diameters (20 mm, 15 mm, 10 mm) are used for high-viscosity materials. Because the dimensions of the geometry are taken into account before reporting the results, an important aspect of the measurement is to have a properly loaded sample that shows no excess or shortage of material. The material must cover the whole geometry area. In order to prevent slippage for the case of fats, and when working with fat disks (explanation below), 60-grit aluminium oxide sand paper is typically attached to both surfaces using an epoxy glue. Nowadays, rheometer companies sell geometries that are serrated or sandblasted. If sand paper is to be used, it is recommended to cut two circles, one slightly larger than the diameter of the geometry for the bottom Peltier Plate and another one the same diameter as the geometry. Instant Krazy® Glue can be used to attach these two circles to both surfaces. To remove the sand paper later on, let it soak in acetone for at least 5 minutes.

**Variables of interest**

In the dynamic mode, the material is subjected to a sinusoidal stress and the resulting sinusoidal strain is measured. Products like margarine and spreads are expected to behave as solids and hold their shape at room temperature, yet be spreadable. They thus exhibit a “yield” value and are considered “plastic”. The material can be characterized by determination of the yield value as well as *G'* and *G"* (or tan*( 𝛿)*).

**Calibration**

Rheometer calibration is carried out by the instrument manufacturer. The user can perform periodic checks to verify the working condition of the rheometer and to set parameters to be used during testing. This is done after turning the equipment on and opening the software. A daily check requires the following: (a) instrument inertia calibration, (b) geometry inertia calibration, (c) mapping or rotational torque calibration or initialization, (d) gap zeroing, (3) bearing friction check and (f) a temperature calibration. All these “checks” are performed without the sample in place.

The "instrument inertia check" must be performed in order to compensate for any acceleration and deceleration of the motor shaft. In a rheometer, the torque output of the motor is composed of the torque required to overcome the instrument inertia and the torque deforming the sample. The instrument's inertia accounts only for the torque associated with the instrument, not with the sample.

The “geometry inertia check” is carried out with the geometry attached to the spindle. This step checks the inertia of the geometry by rotating the shaft with the geometry attached to it. If sand paper is used for the measurement, then the geometry inertia check should be done with the sand paper already in place.

When performing the "mapping check", the instrument checks if there are any variations in behaviour during one revolution of the shaft. To create the map, the software rotates the drive shaft at a fixed speed, monitoring the torque required to maintain this speed through a full 360° of rotation. The "bearing friction check" is done to compensate for any friction that might appear when performing a temperature ramp.

The “Gap”, the distance between the bottom of the geometry and the top of the Peltier plate, might change due to changes in the temperature. When using temperature ramps it is important to compensate for any thermal expansion or contraction by performing a temperature calibration together with a bearing friction correction. The so-called “zeroing the Gap” is not a calibration, but rather a positioning of the geometry to tell the instrument where “0” is. Every time that a geometry is removed or the instrument turned on, the gap needs to be “zeroed”. When zeroing the Gap is selected, the geometry is brought into contact with the Peltier plate. If sand paper is used, zero gap must be checked with the sand paper attached to the base and geometry and Peltier plate. Two modes can be selected to zero the gap: normal force mode or deceleration, which regulates the speed at which the geometry is brought into contact with the bottom Peltier plate. Many users choose the deceleration mode.

## Experimental Procedure

**Creating the Test Program**

The TA AR2000 allows the user to choose between frequency sweep, strain sweep, stress sweep, temperature ramp, temperature sweep, time sweep or manual. The software prompts the user to enter information regarding the experiment to be performed: (a) conditioning, (b) test and (c) post-experiment.

Conditioning of the sample is done prior to the final measurement and requires control of two parameters: the temperature and the normal force. The normal force is the force that the geometry will keep on the sample while performing the measurement. The only purpose is to maintain contact without slippage between the geometry and the material without causing any deformation. Alternatively, the user may choose to keep the gap constant, which means that the geometry is fixed at a constant distance from the Peltier plate. The selection to work under a constant gap is carry out from the instrument window, and not in the conditioning step. If normal force is used, we recommend a value of 4N for fat disks, which are usually 3.2 mm in thickness as described in the sample preparation section (below).

The "test step" indicates which kind of experiment will be carried out, for example, frequency sweep, stress sweep or strain sweep.

In a "frequency sweep" test the frequency is ramped between a minimum and a maximum value. A sweep between 0.1 to 10 Hz is a good first option when dealing with edible fats. In a frequency sweep, the user needs to select one variable to be controlled: torque (µN·m), oscillation stress (Pa), the displacement (rad), or the %strain.

In a “stress sweep” test, the stress is set for a range of values chosen by the user. A typical range for edible fats is 1 to 1000 Pa. A quick test on a sample will allow the user to determine if there is need to work with higher values. The stress sweep is carried out at one particular frequency that the user must specify. This frequency is selected from the LVR region when a frequency sweep was carried out. Nonetheless, a typical value of 1 Hz is used for edible fats.

The “post-experiment” step is performed to prepare the sample holder for the next experiment. The only variable to control here is the temperature.

Once the three steps are set, the user must load the sample and perform the experiment. It is important that all dynamic tests are performed within the region of interest, the LVR of the particular material to study. The LVS usually comprises deformation angles that do not exceed 1° and the strain should be below 0.01% (Marangoni and Wesdorp, 2013).

**Sample Preparation**

One way of studying edible fats is by using disk of fats. This is a challenge because it implies that the sample needs to be “molded” to a particular shape.

Two cases are discussed here: 1- specimens prepared with almost no manipulation, 2- samples molded from the melt into particular shapes.

The first case concern itself with the study of a material “as it is”. Key here is not to destroy the structure that was achieved during the manufacture of the product. This is easy if one has liquid samples, but it becomes a challenge if one likes to study for example a shortening. A useful tool to use is a hollow punch. Once a cylinder of the material is removed from the bulk, slivers of material can be cut using for example two parallel wires. Ideally, two perfect flat surfaces are desired. Trial and error will show the researcher how to cut the sample properly. Another option is to scoop the material using a spatula and deposited onto the desire mold in order to obtain a shape that can be study with a particular geometry. This way of studying the material has it draw back, as it is possible that by pressing down on the material one introduces forces that change the internal structure, the structure that one is trying to characterize.

For the second case, one can use for example PVC molds with three parts: a flat bottom plate, an intermediate template with holes of uniform dimensions and a flat cover plate. This way, the bottom and the intermediate are assemble together and the melted sample poured into the holes before the cover is attached. Three different intermediate plates have been used in our laboratory: (a) holes 20 mm in diameter and 1 mm in height, (b) holes 20 mm in diameter and 3.2 mm in height and (c) holes 10 mm in diameter and 3.2 mm in height. The three plates are screwed in place with butterfly nuts.

Selecting the right intermediate plate is not easy either as if the mold creates a disk that is too thin, the disk will break when the rheometer geometry applies a normal force or when a serrated geometries bites into the disk. If the sample is too thick, only the parts closer to the geometry will be sheared, while the rest of the sample will not experience any shear forces. A good compromise lies somewhere ~3 mm thickness.

The procedure to prepare a solid disk of material using one of the molds requires some simple steps. The first step is to melt the material in the oven to erase all crystal memory. In general, 60°C for 30 minutes or 80°C for 15 minutes is used. The second step requires warming of all materials that will be used to assemble the molds: tools to handle the melted material, strips of either parafilm^{TM®} or aluminium foil and the components of the molds. This will avoid crystallization of the material during the procedure. The third step is the preparation of the base of the mold for which a strip of aluminium foil or parafilm^{TM®} is placed on top of the bottom part of the mold to prevent the fat from sticking to the mold’s surface. For edible fats with high melting point fractions, it is recommended to use aluminium paper, as parafilm^{TM®} melts when in contact with a fat sample above 50°C. The following step requires the insertion of the middle plate on top of the bottom one. The bottom plate screws that easily guide the insertion of the middle plate. The fifth step consists of pouring the molten sample into the mold holes. A strip of parafilm^{TM®} or aluminium foil is placed on top of the molten fat. The intent is to prevent the sample from sticking to the mold. Trial and error will show the user which material is better and whether it is really necessary to use it or not. It is important to avoid air bubbles forming on the surface of the melted fat. Finally, the mold can be closed up by inserting the top plate. Again, the screws attached to the bottom plate are the guides to align the top plate, which is held in place by butterfly nuts. If a particular crystallization temperature is desired, the plastic mold can be placed in an incubator immediately after pouring in the melted fat.

One problem with these molds is that air bubbles may form at the surface of the fat disk (as the fat crystallizes, it contracts and forms air bubbles). These observations prompted us to not use the top plate of the mold for problematic samples. Under these conditions, the fat crystallizes with no bubbles but it creates uneven surfaces that require trimming. The trimming is done with a razor and usually, only the perimeter requires touching up. The amount of stress or strain that might be introduced because of this trimming is unknown.

**Loading the Sample **

Loading of samples is different for soft pastes like partially crystallized fats than for solidified disks. When working with soft fats, start with the geometry far from the sample and load the sample by using a spatula and placing enough material between the geometry and the Peltier plate. As the geometry is brought down towards the Peltier Plate, it reaches contact with the sample. At this point, material will be squeezed by the geometry until it reaches the pre-set gap or Normal force. The material that extends beyond the geometry is then trimmed.

When working with fats that are crystallized into disks, the geometry is raised, a sample disc is placed on the Peltier Plate (sandblasted or with sand paper), and the geometry is then lowered to bring it into contact with the sample. The big decision here is to stop at a fixed gap or at a set normal force. Different results can be obtained depending on whether one sets a distance or a normal force value. One could choose to stop the geometry upon touching, but in our experience, this leads to “slip”, which invalidates results. The top geometry must be in mechanical contact with the sample. Too high a normal force may lead to sample breakage, also invalidating results. The sweet spot must be determined experimentally.

## Two Case Studies

**Fully Hydrogenated Soybean Oil**

Results for a study of fully hydrogenated soybean oil (FHSO) (Bunge, Toronto, Ontario, Canada) using a dynamic method are presented in this section. Fat disks were crystallized using molds 20 mm in diameter and 3.2 mm in height. Two measurements were carried out: a frequency sweep and a shear stress sweep. FHSO was tested by making disks of 100% FHSO as well as disks containing 70% FHSO and 30% canola oil. A normal force of 4N was selected in order to close the rheometer gap. The temperature was kept constant at 20°C.

A frequency sweep from 1 to 250 rad/s was performed to determine a suitable frequency to carry out the stress sweep. The behaviour of *G*' and *G"* for both samples is shown in Fig. 6.

Both *G*' and *G*" have higher values for the sample containing oil than for the 100% FHSO sample. This result does not initially make sense since the presence of oil should decrease the elastic modulus of the sample. However, powder X-ray diffraction studies showed that the 100% FHSO sample is in the alpha polymorphic form while 70% FHSO sample is in the beta polymorphic form. Any of the frequencies shown in Fig. 6 could be chosen to perform the stress sweep as the value of *G*' is constant in the region studied, showing a wide LVR.

The range of frequencies analysed in these experiments shows that the behaviours of *G *'and *G*" are not frequency-dependent. A frequency of 1 Hz or 6.28 rad/s was therefore chosen, as it has become the typical starting value for researches using fat disks, yet the user can work at any frequency in the range where it remains constant. The stress sweep measurement was then carried out in the region from 1 to 10000 Pa at a frequency of 1 Hz. Results of the stress sweep are displayed in Fig. 7

Storage (*G*') and loss (*G"*) moduli are taken from the LVR region. The yield stress can be obtained from the point at which the slope starts to change for *G*', *G*" and tan(𝛿) (Fig. 7). The breaking point is the point that causes a permanent deformation in the sample, and it usually occurred before the three curves (cross (Fig. 7A). In both samples, *G*' is higher than *G*" in the LVR region, which indicates that the material is more elastic than viscous. The sample containing oil is able to withstand a higher stress before yielding and breaking.

**Tristearin-Triolein Blends**

Another case study is presented in Table 1, which shows the results for a system composed of interesterified and non-interesterified tristearin (SSS)-triolein (OOO) blends crystallized and stored at 20°C and 30°C (Rogers *et al*., 2008). Fat disks were prepared following the sample preparation described above. Prepared fat disks were stored sample at 20°C and 30°C for 24 hours before measurements were carried out. A stress sweep was performed at 1 Hz. Results displayed in Table 1 show that *G*' is higher for the interesterified sample at 20°C than at 30°C. *****G'* increases as the percentage of tristearin solids increases and also upon interesterification. This is to show that the processing conditions and the final temperature help to achieve different values for the storage modulus.

## Rheological-Fractal Model

For the last 50 years, efforts had been made to try to correlate the storage modulus, *G*' to the solid fat fractions. The first theoretical model was presented by van den Tempel (Van den Tempel 1961). Van den Tempel postulated that edible fats contain flocculated solid particles of colloidal dimensions embedded in a liquid phase. He proposed a linear relation between *G*' and the solid fat fraction, which was later not corroborated by experiments. A modification to the model was later introduced (van den Tempel, 1979) without solving all the problems. An advance was made when Shih, et al. (1990), outlined, for edible fats, the scaling theory of colloidal gels proposed by Brown and Ball (1985), which was based on a power-law relationship between the shear elastic modulus and the solid volume fraction. Studies using light scattering (Vreeker *et al*., 1992) and microscopy showed that edible fats were fractal in nature. This allowed to farther modified the model introducing an exponent in the power law that took into account the observed fractality (Narine and Marangoni, 1999a, 1999b; Marangoni, 2000, 2002; Marangoni and Rogers, 2003).

The rheological-fractal model currently describes spherical clusters of solid edible interacting exclusively via van der Waals forces. The model (Marangoni and Rousseau, 1996; Narine and Marangoni, 1999a; Marangoni, 2000, 2002; Marangoni and Rogers, 2003) proposes that the storage modulus ', is related to the Hamaker coefficient, *A _{H}* and the solid fat fraction Φ by Eq. 14, 15:

^{1⁄(d-DR)}

**(Eq. 14)**

_{H}/(6 π a γ d

_{0}

^{2})

**(Eq. 15)**

Where D_{R} = 3 —1/*slope* and the slope is the one obtained when plotting In(*G*') against In(Φ). To achieve this, samples made with different proportions of Φ are made and *G*' for each of them measured. α is the primary crystal diameter, d_{o} is the distance between primary crystals. An on going problem is the determination of the distance between the primary crystals.

## Remarks

Developing faster, convenient, “fool-proof” and more reliable equipment is the ultimate goal of many equipment manufacturers. But, if the operator neglects to understand the instrument’s capabilities, measurement principles and limitations, all the manufacturer’s efforts will be lost. It is imperative that every operator understands not only how to run a sample, but also why the equipment was designed the way it was. This understanding will help the operator make educated guesses regarding the variables that need to be kept fixed and the ones that can be changed.

In this section, we showed how rheology can successfully be used to help characterize the microstructure and flow properties of fats and oils. Using dynamic methods to, for example, obtain the storage module can help quantify and characterize edible fat systems. The use of models, like the rheological-fractal model, contributes to a better understanding of the microstructure of systems.

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