In this thesis, the interest is on the behaviour of the bulk paste and how it behaves under different conditions. Rheology is the primary characterisation technique for examining the flow properties of bulk material. As the key removal technique is fluid flow, it is useful to understand the flow properties of the toothpaste and examine what influence this has on the cleaning behaviour of the toothpaste(s). Rheology is used to gain understanding on the internal structure of a material and to study how the material responds to stress and shear rate as discussed by Steffe (1996). In oscillatory rheometry, the substance to be tested is sheared between two surfaces. The resistance to movement is measured and is analogous to any deformation in the fluid. There are a number of parameters which describe characteristics of the bulk behaviour of the paste(s) including yield stress, rheological models and the viscoelastic properties of the paste(s).
2.3.1. Yield stress
Toothpaste is classically described as having a yield stress [Steffe (1996)] as toothpaste does not simply flow out of the toothpaste tube when turned upside down. A force (squeezing) is required to enable the toothpaste to flow, this property is known as the yield stress. Yield stress is the amount of stress which is applied to a sample before it is seen to flow, typically yield stress fluids do not start flowing slowly. It is impossible to define yield stress unambiguously as given enough time or accurate enough measuring equipment any material could be seen to flow [Steff (1996)]. Pan et al (2004) defines yield stress as an empirical constant which depends on the experimental conditions and material and that it is an „apparent‟ instead of a „true‟ property of a fluid. The yield stress is a function of the microstructure of the fluid that resists large rearrangements. A minimum shear is applied to the fluid after the resistance to shear has been overcome, and then the material can flow. Yield stress is a controversial topic and has been discussed in a letter by Coussot (2002). Coussot (2002) states that when a material is submitted to flow the microstructure is partially destroyed, Coussot (2002) found that for most systems at rest the structure reforms or evolves spontaneously and the system is said to age. The yield stress property is captured in a number of rheological models.
2.3.2. Rheological models
Gunasekaran & Mehmet Ak (2000) discussed how an ideal solid material responds to an applied load by deforming finitely and recovering that deformation upon removal of the load. This is described as elastic behaviour. Material will not recover from its deformation when the load is removed and there is complete loss of energy as all the energy supplied during deformation is dissipated as heat and obeys Newtons law. However, real materials have characteristics which are in between the extremes and rheology can be used to characterise the extent to which this is the case. Some of the most common materials, for example foods, are shear thinning materials. Shear thinning materials become less viscous as the shear stress is applied, hence it loses some of its structural order.
The toothpaste is expected to be shear thinning and have an apparent yield stress, the Herschel-Bulkley model fits these behaviours, (TA Model Note (2007)) as given in equation 2.3:
n
k
y (2. 3)
where σy = yield stress, σ = stress, k = constant, γ = shear rate, n = rate index.
Cross and Williamson models as given in equation 2.4 and equation 2.5 respectively, also have shear thinning and apparent yield stress behaviour. They contain more parameters than the Herschel-Bulkley model.
Cross Model: m dt d K 0 (2. 4) Williamson Model: 1 1 0 n dt d K (2. 5)
where η= viscosity (Pa.s), K=Consistency factor, γ=shear rate (1/s), t=time (s), m=rate index, n=rate index.
These models are a useful tool in describing the behaviour of toothpaste mathematically. The better a model can predict the toothpaste behaviour, the better the interparticle forces can be understood.
2.3.3. Viscoelastic properties of toothpaste
A sample can be further interrogated through oscillatory shear experiments where the sample is subjected to different frequencies and the response measured in terms of its elastic and viscous modulus. Gunasekaran & Mehmet Ak (2000) describe the dynamic tests required to characterise a viscoelastic material, the first stage of this is to determine the Linear Viscoelastic Region (LVR) using a SAOS (Small Amplitude applied Oscillatory Shear) test using very small stresses or strains to identify whether (and in what shear range) the material response is linear so that the stress is proportional to the applied strain. The linear region is
the region which has the maximum stress that can be applied to the structure without the structure being destroyed.
The shear elastic modulus (G‟) and the shear viscous modulus (G‟‟) are the frequency dependant functions. The elastic modulus is a measure of stored energy (storage modulus) which describes the molecular events of an elastic nature. The viscous modulus is a measure of the energy dissipated as heat per cycle of deformation per unit volume (loss modulus) and describes the molecular events of viscous nature. Gunasekaran & Mehmet Ak (2000) show that the stress measured is a combination of the elastic and viscous components, see equation 2.6. ) cos( ) ( ' ' ) sin( ) ( ' ) (t 0G t 0G t (2. 6)
Elasticity in toothpaste has been observed by Prencipe et al (1995) who stated that “highly elastic structures are less subject to phase separation during the aging process” and that the toothpaste “recover(s) more quickly after a stress is imposed on the toothpaste”. This is a desirable property which means that the toothpaste „stands up‟ on the brush and doesn‟t flow off it.
Gunasekaran & Mehmet Ak (2000) refer to weak gels where the elastic modulus is greater than the viscous modulus (G‟>G‟‟), where both the storage and loss modulus are largely independent of frequency and the linear viscoelatic strain limit is small, with a shear rate of less than 0.05. These weak gels are described by Gunasekaran & Mehmet Ak (2000) as having an entangled network system. Toothpaste is known to exist as a three-phase system; the continuous phase; the dispersed phase and the solid phase. The continuous phase suspends the solid phase in a gel using binders present in the dispersed phase [Prencipe et al (1995)]
and so an entangled network would correlate with the known properties of toothpaste.