Ideally, time-dependent materials are considered to be inelastic with a viscosity function which depends on time. The response of the sub-stance to stress is instantaneous and the time-dependent behavior is due to changes in the structure of the material itself. In contrast, time effects found in viscoelastic materials arise because the response of stress to applied strain is not instantaneous and not associated with a structural change in the material. Also, the time scale of thixotropy may be quite different than the time scale associated with viscoelasticity:
The most dramatic effects are usually observed in situations involving short process times. Note too, that real materials may be both time-dependent and viscoelastic!
reduced viscosity= ηred=ηsp
C inherent viscosity= ηinh=lnηrel
C intrinsic viscosity= ηint=
ηsp
C
C→0
C
Figure 1.17. Time-dependent behavior of fluids.
Separate terminology has been developed to describe fluids with time-dependent characteristics. Thixotropic and rheopectic materials exhibit, respectively, decreasing and increasing shear stress (and apparent viscosity) over time at a fixed rate of shear (Fig. 1.17). In other words, thixotropy is dependent thinning and rheopexy is time-dependent thickening. Both phenomena may be irreversible, reversible or partially reversible. There is general agreement that the term
"thixotropy" refers to the time-dependent decrease in viscosity, due to shearing, and the subsequent recovery of viscosity when shearing is removed (Mewis, 1979). Irreversible thixotropy, called rheomalaxis or rheodestruction, is common in food products and may be a factor in evaluating yield stress as well as the general flow behavior of a material.
Anti-thixotropy and negative thixotropy are synonyms for rheopexy.
Thixotropy in many fluid foods may be described in terms of the sol-gel transition phenomenon. This terminology could apply, for example, to starch-thickened baby food or yogurt. After being man-ufactured, and placed in a container, these foods slowly develop a three dimensional network and may be described as gels. When subjected to shear (by standard rheological testing or mixing with a spoon), structure is broken down and the materials reach a minimum thickness where
Thixotropic
Time-Independent
Rheopectic Shear Stress, Pa
Time at Constant Shear Rate, s Time-Dependent Behavior
Figure 1.18. Thixotropic behavior observed in torque decay curves.
they exist in the sol state. In foods that show reversibility, the network is rebuilt and the gel state reobtained. Irreversible materials remain in the sol state.
The range of thixotropic behavior is illustrated in Fig. 1.18. Sub-jected to a constant shear rate, the shear stress will decay over time.
During the rest period the material may completely recover, partially recover or not recover any of its original structure leading to a high, medium, or low torque response in the sample. Rotational viscometers have proven to be very useful in evaluating time-dependent fluid behavior because (unlike tube viscometers) they easily allow materials to be subjected to alternate periods of shear and rest.
Step (or linear) changes in shear rate may also be carried out sequentially with the resulting shear stress observed between steps.
Typical results are depicted in Fig. 1.19. Actual curve segments (such as 1-2 and 3-4) depend on the relative contribution of structural breakdown and buildup in the substance. Plotting shear stress versus shear rate for the increasing and decreasing shear rate values can be used to generate hysteresis loops (a difference in the up and down curves) for the material. The area between the curves depends on the time-dependent nature of the substance: it is zero for a time-intime-dependent
0
Stress
0
time Rest Period
Shear Rate
Complete Recovery Partial Recovery No Recovery
Evidence of Thixotropy in Torque Decay Curves
fluid. This information may be valuable in comparing different materials, but it is somewhat subjective because different step change periods may lead to different hysteresis loops. Similar information can be generated by subjecting materials to step (or linear) changes in shear stress and observing the resulting changes in shear rates.
Figure 1.19. Thixotropic behavior observed from step changes in shear rate.
Torque decay data (like that given for a problem in mixer viscometry described in Example Problem 3.8.22) may be used to model irreversible thixotropy by adding a structural decay parameter to the Herschel-Bulkley model to account for breakdown (Tiu and Boger, 1974):
[1.36]
where , the structural parameter, is a function of time. before the onset of shearing and equals an equilibrium value ( ) obtained after complete breakdown from shearing. The decay of the structural parameter with time may be assumed to obey a second order equation:
[1.37]
time
Shear StressShear Rate
1
2 3 4
Step Changes in Shear Rate
σ =f(λ,˙γ) = λ(σo+K(γ)˙n)
λ λ =1
λ λe
dλ
dt = −k1(λ − λe)2 for λ > λe
where is a rate constant that is a function of shear rate. Then, the entire model is completely determined by five parameters: , and . and are determined under initial shearing conditions when and . In other words, they are determined from the initial shear stress in the material, observed at the beginning of a test, for each shear rate considered.
and are expressed in terms of the apparent viscosity ( ) to find . Equating the rheological model (Eq. [1.36]) to the definition of apparent viscosity (which in this case is a function of both shear rate and the time-dependent apparent viscosity) yields an expression for :
[1.38]
Eq. [1.38] is valid for all values of including at , the equilibrium value of the apparent viscosity. Differentiating with respect to time, at a constant shear rate, gives
[1.39]
Using the definition of , Eq. [1.37] and [1.39] may be combined yielding
[1.40]
Considering the definition of given by Eq. [1.38], this may be rewritten as
[1.43]
where
[1.44]
Integrating Eq. [1.43] gives
[1.45]
so
[1.46]
where is the initial value of the apparent viscosity calculated from the initial ( and ) shear stress and shear rate.
Using Eq. [1.46], a plot versus , at a particular shear rate, is made to obtain . This is done at numerous shear rates and the resulting information is used to determine the relation between and and, from that, the relation between and . This is the final infor-mation required to completely specify the mathematical model given by Eq. [1.36] and [1.37].
The above approach has been used to describe the behavior of mayonnaise (Tiu and Boger, 1974), baby food (Ford and Steffe, 1986), and buttermilk (Butler and McNulty, 1995). More complex models which include terms for the recovery of structure are also available (Cheng, 1973; Ferguson and Kemblowski, 1991). Numerous rheological models to describe time-dependent behavior have been summarized by Holdsworth (1993).