Small deformation rheology

For a plastic fat, the yield stress was defined as the stress at the limit of linearity in a small deformation rheological test. Agreement between theory and experiment was found to be good. 

When using small deformation rheology there are several useful parameters that may be obtained to describe a material the complex modulus (G ), storage modulus (G ), loss modulus (G") and the tangent of the phase shift or phase angle (tan 5). These values must be taken from within the LVR, and are obtained using a dynamic oscillatory rheometer (Rao 1999). Outside the LVR, important information may be obtained such as the yield stress and yield strain. 

In practice, the fractal dunensions of fat crystal networks, are measured by determining the shear storage modulus, G, of the fat crystal networks within the LVR measured by small deformation rheology. The fractal dimension of the fat crystal network may be calculated from the slope of the curve as > = 3-1/slope of logarithm of G at different SFC, O. 

Chronakis, 1. S., Kasapis, S., and Richardson, R. K. 1996a. Small deformation rheological properties of maltodextrin-milk protein systems. Carbohydr. Polym. 29 137-148. 

The textural properties of a fat are influenced by all levels of structure, particularly microstructure. The microstmcture includes the spatial distribution of mass, particle size, interparticle separation distance, particle shape, and interparticle interaction forces (49-51). Methods that can be used for the characterization of microstructure in fat systems include, among others, small deformation rheology and polarized light microscopy, employing a fractal approach (49-51). 

Fractal Dimension Evaluation using Small Deformation Rheology (Dr) To determine for a particular system by rheological methods, the shear storage modulus (G ) must be measured at different solids volume fractions. The solids volume fraction, which is equal to the SEC/100, is varied by diluting the fat with an inert solvent that will not cocrystallize, or otherwise alter the behavior of the system. Canola oil has been used as a suitable AMF diluent for these purposes. A log-log plot of G versus <1) yields a slope p from which Dj can be... 

Figure 16. Time profile of an applied sinusoidal stress wave and the corresponding resulting sinusoidal strain wave as they apply to small deformation rheological testing.

The data from small deformation rheology indicate that at higher cooling rates, the fat samples are more solid-like in nature. Conversely, the low cooling rate of 0.1°C/min results in a softer, less elastic system. 

Figure 5.12 Effect of i-carrageenan added after emulsification on small-deformation rheology of fine BSA-stabilized emulsions (2.7 w/% protein, 40 vol% n-tetradecane, dsi = 0.55 pm, pH (5). Complex shear modulus G at I Hz and 30 C is plotted against polysaccharide concentration c (Reproduced by permission from ref. 107)...

In a later study [56], the effect of gas volume fraction (foam rheology was investigated. Two models were considered one in which the liquid was confined to the Plateau borders, with thin films of negligible thickness and the second, which involves a finite (strain-dependent) film thickness. For small deformations, no differences were observed in the stress/strain results for the two cases. This was attributed to the film thickness being very much smaller than the cell size. Thus, it was possible to neglect the effect of finite film thickness on stress/strain behaviour, for small strains. 

Markovitz,H. Small deformations superimposed-cm steady viscometric flows. In Onogi,S. (Ed.) Proc. 5th Intemat. Cong Rheology, Vol. I, pp. 499-510. Maryland University Park Press 1970. 

Most of the methods used to characterize the rheological behavior of butter are empirical and attempt to imitate certain sensory perceptions. They typically involve penetrometry, extrusion or sectility tests (Prentice, 1972). In these tests, the structure of the material is destroyed in order to probe its response to an applied stress or deformation. These methods mostly serve a quality control function. Their results provide an index of consistency to adjust milk-blending operations or to regulate a step in the butter-making process. While the results have practical significance, they often have no theoretical basis. Therefore, attempts have also been made to study the intrinsic properties of plastic fats. In many such cases, small deformation tests, in which the structure of the sample remains intact have been used to probe milk fat rheology. 

Rohm, H., Weidinger, K.H. 1993. Rheological behaviour of butter at small deformations. J. Text. Stud. 24, 157-172. 

The conclusion is that Lodge s rheological constitutive equation results in relationships between steady shear and oscillatory experiments. The limits y0 0 (i.e. small deformation amplitudes in oscillatory flow) and q >0 (i.e. small shear rates) do not come from Lodge s equation but they are in agreement with practice. These interrelations between sinusoidal shear deformations and steady shear flow are called the relationships of Coleman and Markovitz. 

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

For a viscoelastic solid (like an organogel), any rheological description should give a constant finite elastic modulus and infinite viscosity at zero frequency or long times. The situation is somewhat comparable to that of a cross-linked network [2. The equilibrium shear modulus for small deformations is proportional... 

Rohm, H., and Weidinger, K.H. (1993). Rheological behavior at small deformations. Journal of Texture Studies. 24 157-172. 

Genovese, D. B., Acquarone, V. M., Youn, K.-S., and Rao, M. A. 2004. Influence of fructose and sucrose on small and large deformation rheological behavior of heated Amioca starch dispersions. Food Sci. Technol. Int. 10(1) 51-57. 

J. F. Steffe, in Rheological Behaviour of Butter at Small Deformations, 2nd ed.. Freeman Press, East Lansing, Michigan, 1996, pp. 294—349. 

In real food polymers, a distinction can be made between a viscoelastic solid, which contains some cross-links that are permanent, and a viscoelastic liquid, where, under the influence of stress, the relative movement of whole molecules will be observed. As shown in Figure 8.9, in the case of a viscoelastic solid, after application of the stress, the strain will eventually reach a constant value, and upon removal of the stress, the strain will finally return to the remaining value of food primary energy, which was not entirely dissipated. For a viscoelastic liquid, a permanent deformation will remain after removal of the stress. In the stress relaxation area, the deformation value will decay to zero for a viscoelastic liquid, whereas for a solid, it will reach a constant, nonzero value. It can also be seen as either a decreased value to the zero or a constant, nonzero value, as pointed out by the dashed line. Both values characterize the rheology parameters of foods under certain conditions. One of the reasons for this is that the factors of time-dependent foods are not necessarily related to their elastic modulus. This can be explained by the series of small deformations without rupture, which are dependent in different ways and are based on the primary molecular microstructure of foods. The time required for the stress to relax to a definite fraction of its initial value is the relaxation time. 

Since AL is much larger for the rubber than for the plastic, from equation (1-2) it is clear that the modulus of the rubber is much lower than the modulus of the plastic. Thus the particular modulus defined in equation (1-2) specifies the resistance of a material to elongation at small deformations and is called the Young s modulus. It is normally given the symbol E. (See www.rheology.org for suggestions on standard nomenclature for viscoelastic quantities.)... 

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