The Rheology Handbook. Thomas Mezger
with decreasing, and (2) with increasing structural strength when shearing
Figure 3.44: Flow curves obtained when controlling the shear stress, showing a hysteresis area: (1) with decreasing, and (2) with increasing structural strength when shearing
3.4.2.2.9Example 2: Preset of the shear stress
1st interval (shear stress ramp upwards, in t = 120 s): with τ = 0 to 2000 Pa
2nd interval (high-shear phase, for t = 60 s): at τ = 2000 Pa = const
3rd interval (shear stress ramp downwards, in t = 120 s): with τ = 2000 to 0 Pa
3.4.2.2.10Measuring result
1 With CSR: Flow curves τ( γ ̇ ) showing a hysteresis area, see Figure 3.43
2 With CSS: Flow curves γ ̇ (τ) showing a hysteresis area, see Figure 3.44
The area between the upward and downward curves in Figures 3.43 and 3.44 is called hysteresis area. Hysteros is a Greek word meaning later, and hysteresis literally is the dependence of a (physical) state on previous states [3.65]. In this case, we can interpret this term as a time-dependent loop or a loop which is influenced by time-dependent behavior. In the past, the terms thixotropic area or rheopectic area, respectively, were used. The thixotropic value was determined by the following procedure.
1 When presetting the shear rate, determination of the difference of the first area between the upward curve and the γ ̇ -axis, and the second area between the downward curve and the γ ̇ -axis: Samples for which a positive value of resulting area is obtained are termed thixotropic (no. 1), and those with a negative value are called rheopectic (no. 2).
2 When presetting the shear stress, evaluation of the difference of the first area between the downward curve and the τ-axis, and the second area between the upward curve and the τ-axis: Samples for which a positive area value is obtained are referred to as thixotropic (no. 1), and those with a negative value are named rheopectic (no. 2).
Note 1: Hysteresis area and shearing power
The shearing power which corresponds to the thixotropic structural decomposition, can be calculated as the product of the total amount of the area between the two measured flow curves and the sheared volume of the sample [3.69]:
P = [(τ ⋅ γ ̇ )1 – (τ ⋅ γ ̇ )2] ⋅ V
with τ [Pa], γ ̇ [s-1] and V [m3]. The resulting unit is 1 (Pa ⋅ m3) / s = 1 Nm/s = 1 J/s
It is, however, very optimistic to assume that all parts of the whole sample are sheared homogeneously, and therefore at a constant shear load or shearing power.
Comment: No undisturbed structure regeneration
Using the hysteresis area method, flow behavior is only determined in a state of motion, and thus, only in the phase of structural decomposition. This method does not reveal any information about structural regeneration at rest, since a state of rest is not part of the test method at all. This is why in current literature the analyzed hysteresis area no longer is referred to as the thixotropic area or rheopectic area, respectively. However, in most cases it is just the phase of regeneration which is important for practical users if leveling and sagging behavior after an application process of a coating has to be evaluated. As a consequence, it is therefore better to use the step test here, as explained above in Chapter 3.4.2.2a to determine thixotropic behavior [3.80].
Note 2: Hysteresis area, and different structural changes for ramps with increasing and decreasing shear rates
Suspensions may show different morphologies, not only depending on the shear rate, but also on the direction of the shear rate ramp. Example: (1) Ramp with increasing shear rates: Initially at rest or at low shear rates, the sample showed a homogeneous structure, and therefore, relatively high viscosity values. At intermediate shear rates a flocculated structure was produced, thus, loosely bound flocs and water-rich voids reduced the overall viscosity. (2) Ramp with decreasing shear rates: After reaching again the range of low shear rates, the flocculated structure differs from the initial structure showing relatively lower viscosity values now.
Example: Dependent on the shear rate range, micro- and nano-fibrillated cellulose materials (MFC, NFC), displayed alternating hysteresis areas with on the one hand decreasing viscosity values (named positive hysteresis) and on the other hand increasing viscosity values (named negative hysteresis). [3.86]
3.4.2.2.11c) Very simple evaluation methods (for evaluating thixotropic behavior)
1) Time-dependent viscosity ratio or “thixotropy index” (using a single test interval only)
Some users evaluate thixotropic behavior by the following simple testing and analysis method: with a single test interval, presetting a constant medium or high rotational speed n = const (or shear rate). Afterwards the thixotropy index (TI) is calculated as follows
TI = η1(t1) / η2(t2)
with the time points t1 and t2 [s], (e. g., t1 = 30 s, and t2 = 600 s). For flow behavior independent of time TI = 1, for time-dependent shear-thinning TI > 1, and for time-dependent shear-thickening TI < 1.
Comment: Here, the term thixotropy index is misleading since this ratio quantifies time-dependent structural decomposition of a material only. Thixotropic behavior, however, can only be quantified if – directly after the break of a material’s superstructure – also the subsequent time-dependent structural recovery under low-shear condition is evaluated (to TI, see also Note 3 in Chapter 3.4.2.2a, and Note 2 in Chapter 3.3.2). Therefore, instead of TI, this ratio should better be called time-dependent viscosity ratio under a constant shear load, or similar.
2) Bingham build-up (BBU) and rate of build-up (RBU) after a 20-minute gelation test consisting of two test intervals: first high, then low shear load [3.7].
Some users perform the following simple test and analysis method consisting of two intervals. In the first part, preset is a constantly high rotational speed nH [min-1] for a period of t10 = 10 min, and in the second part a constantly low speed nL for another 10min = t20 (e. g. for ceramic suspensions, with nL = nH /10, for example, at nH = 100 min-1 and at nL = 10 min-1). Please be aware that these η-values are relative viscosity values if the test is performed using a spindle (which is a relative measuring system; see also Chapter 10.6.2). Here, instead of the shear stress often is used dial reading DR (which is the relative torque value Mrel in %), and the viscosity values are calculated then simply as η = DR/n (with the rotational speed n in min-1). Usually here, all units are ignored.
Bingham build-up (BBU) indicates the change of the relative viscosity values between the end of the second, low-shear interval and the first, high-shear interval.
Calculation: BBU = ηL (nL, t20) – ηH (nH, t10) = (DRL/nL) – (DRH/nH).
Example: with nH = 100, nL = 10, DRH = 50, DRL = 40,
then: BBU = (40/10) – (50/100) = 4 – 0.5 = 3.5
Rate of build-up (RBU) is the partial change over the first two minutes in the second, low-shear interval related to the total change over the full ten minutes in this interval.
Calculation: RBU = [η(nL, t12) – η(nH, t10)] / [η(nL, t20) – η(nH, t10)]
= [(DRL, 12/nL) – (DRH/nH)] / [(DRL/nL) – (DRH/nH)]
with time