## Part 2. Viscoelasticity

The sauce is separating, the butter is melting and the marmalade is setting in the process. These are examples of rheological changes which are explained best using viscoelastic terms.

In the Rheology for beginners, part 1, viscosity was explained and how it is used. Unfortunately viscosity can not describe what is occurring when something is melting or setting because the viscosity is infinite for a solid material. Then we have to use dynamic magnitudes such as modulus and phase angel to describe the consistency and character.

If we start with an example which many of us have played with – bouncing putty – it is a typical material which can not be described fully by its viscosity or its elasticity. Bouncing putty can be formed and slowly pulled into long threads. If it is formed to a ball it will bounce like a bouncing ball. Bouncing putty together with many foods, for example marmalade, mayonnaise and spreads, are materials which are both viscous and elastic, viscoelastic. For these the complex modulus is needed, which for shear is represented by G*. It can be divided into two parts, the storage modulus G’ which describes the elastic properties, and G”, the loss modulus which describes the viscous properties. What is de facto a modulus?

There are a number of different modules and for an elastic material you could say that the modulus describes the stiffness of a material. The spring constant is a typical modulus. It describes how much force is needed to stretch a spring to a certain distance, with other words the force divided with the length. The same conditions can be applied on the complex modulus but instead force per area divided with dimensionless deformation is used, see the formula box. The modulus can be measured in different ways and one of the most commonly used is shown in the figure below. A viscoelastic material is poured into a cup and a concentrical cylinder (called ”bob”) is lowered into the material. The cup is oscillating in a sinus like movement and the force which the material is transferring is measured with the concentric cylinder. The force will also be sinus like with the same frequency as the cup, but displaced in comparison to the movement. The displacement is measured as a phase angel and the variations correspond to the character of the material. A totally elastic material, such as steel, has no displacement at all, and the phase angel δ=0° while a liquid has maximal displacement and δ=90°. The onset deformation, the measured force and displacement are used to describe the storage modulus and the loss modulus.

Sketch of a rheometer for viscoelastic measurements. Click for an animation.

Viscoelastic measurements at different frequencies can be used to describe the character for a material in the same way as an optical spectrum is used for spectrophotometers. A mechanic spectrum is shown for bouncing putty in figure 2. The modules vary dramatically with frequency, which very well describes how the bouncing putty is experienced during playing. Low frequencies correspond to when long threads are slowly pulled from the bouncing putty. The loss modulus which describes the viscous properties is then much higher than the storage modulus – the bouncing putty behaves like a viscous liquid. At high frequencies the storage modulus is much higher than the loss modulus so the bouncing putty has an elastic behaviour and bounces. The bounce is quick and corresponds to high frequency.

A mechanical spectrum for bouncing putty. At low frequencies G” dominates and the bouncing putty behaves like a viscous liquid. At high frequencies G’ dominates and the bouncing putty behaves like an elastic bouncing ball.

Viscoelastic measurements are also useful for characterising changes like melting, crystallisation, and gel formation. The modules are then measured at a certain frequency while other parameters are changed, such as temperature or pH. If we use boiling an egg as an example the liquid egg white will turn into solid material when the temperature is increased.  Exactly the same thing can be performed in our rheometer in figure 1. The egg white is poured into the cup and the complex modulus is measured while the temperature is increased. The albumin in the egg white will aggregate and form a gel in the measuring cup when the temperature is high enough. The same thing is valid for many more proteins, figure 3 shows how the β-lactoglobulin from whey protein forms a gel when the temperature is increased. (Note that only G’ and d is shown. Only two parameters are needed to describe a viscoelastic material, which ones chosen are equal.) At low temperature the β-lactoglobulin solution is a liquid which is explained by the high phase angel. The liquid has a low viscosity why the measured signal becomes noisy.  When the temperature is increasing the protein aggregates and G’ increases simultaneously as d decreases, which means that the characteristics for the solution changes from liquid towards a more elastic material. This happens in two steps and at 75°C, G’ is dramatically increasing when the protein denaturates.

Gel formation of the protein β-lactoglobulin observed using viscoelastic measurements at 1 Hz. (Stading and Hermansson, 1990).

One advantage with viscoelastic measurements is the extremely small deformation of the sample, only as much as required for performing a measurement and yet so little as the material does not change. Because of this it is possible to perform repeated measurements on the same sample and follow the changes occurring in the sample. The applied deformation must be tested utterly thorough so the material does not change during the measures.  For the measurement shown in figure 3 a deformation g=10-3 was used which corresponds to a maximum movement of 8 mm. If the movement is increased to 30 mm, which still is a very small deformation, the first structure formation will be destroyed and the final storage modulus will be six times lower! It is extremely important to use a deformation which is small enough.

Bouncing putty and pure protein solutions may seem far from real food, but the same methods are used to characterise the processes when the sauce is separating or when the butter is melting.  Pure and simple systems are easier to use as examples, but a mechanic spectrum for marmalade or yoghurt is as interesting as for bouncing putty.

The author demonstrats flow of bouncing putty

Formulas