(τ0 / G2) ⋅ [1 – (1/e)]
thus: γ(Λ) = γmax – (τ0 / G1) – 0.632 ⋅ (τ0 / G2)
i. e., S1 is fully recoiled; and up to this time point, S2 and therefore also D2 is reset by 63.2 %.
D3 however, is still fully deflected as it was at the end of the creep phase.
6.3.4Definition of terms
6.3.4.1Zero-shear viscosity
Zero-shear viscosity η0 is one of the most important rheological parameters used in polymer industry. The value of η0 is determined at the end of the creep phase as
η0 = τ0 / γ ̇ 3 = τ0 / tanβ
Unit: [Pas]; with the constant shear rate γ ̇ 3 = Δγ3 / (t2 – t1) = (γ3 – γ2) / (t2 – t1) = tanβ which is independent of time (see Figure 6.8); γ ̇ 3 is often referred to as γ ̇ 0.
The value of zero-shear viscosity corresponds to the behavior of the dashpot D3 in the Burgers model, resulting in a “creep rate” value, i. e. in a very low rate of deformation or shear rate. In order to avoid exceeding the limit of the LVE deformation range between the time points t1 and t2, the total deformation value should not increase too much in this period (see also Chapter 6.2.1: values of γmax).
6.3.4.1.1a) Determination of the value of zero-shear viscosity via creep curves
1 Manual or visual determination, respectively
In practice, the aim is to produce a creep curve showing a constant slope at the end of the creep phase. In order to get a useful basis for the analysis, as a rule of thumb, at least 10 % of the measuring points of the entire creep curve should occur within this range of steady-state flow. Taking a ruler, it is useful to draw a straight line in this test interval to facilitate visual evaluation. The following results are obtained from this sector of the curve: The change in deformation Δγ read-off on the γ-axis, and the corresponding period of time Δt read-off on the time axis. The first calculation step is: Δγ /Δt; this is the deformation change in the corresponding period of time, or time-dependent rate of deformation (shear rate) γ ̇ 0. The second calculation step (see Figure 6.8):
Equation 6.3
η0 = τ0 / (Δγ /Δt) = τ0 / [Δγ3 / (t2 – t1)] = τ0 / [(γ3 – γ2) / (t2 – t1)] = τ0 / γ ̇ 0
1 Automatic determination using a software analysis program
First of all, the user defines the part of the final sector of the creep curve to be used for the characterization of steady-state behavior, i. e. when dγ/dt = const is reached. Like above, as a useful rule of thumb at least 10 % of the measuring points of the whole creep curve should occur within this steady-state sector to outline the corresponding curve interval. Then, the straight analysis line is fitted in this curve sector by the software program, finally determining also the value of η0.
6.3.4.1.2b) Zero-shear viscosity and average molar mass
Using the value of zero-shear viscosity η0 for uncrosslinked polymers, a relative specification may be given for the average molar mass M, since the following proportionality holds:
Equation 6.4η0 ~ M3.4
Therefore, polymers with a higher molar mass are showing higher η0-values (see also Chapter 3.3.2.1a3 with Equation 3.3). For non-flowing materials, e. g. for chemically crosslinked elastomers such as crosslinked rubbers, the calculated η0-values obtained would approach towards infinity. Therefore of course, it is not useful to determine any whatsoever η0-value for these kinds of materials, as it senseless for practical users to specify any viscosity value for a solid.
6.3.4.1.3c) Comparison: different methods to determine η0
At a selected temperature, the value of zero-shear viscosity is a material constant which can be determined using the following rheological test methods [6.6]:
1 Rotational tests: η0 as the limiting value of the shear viscosity function η( γ ̇ ) “at rest”, i. e. for γ ̇ → 0 (see Chapter 3.3.2.1a, Equation 3.1)
2 Frequency sweeps (oscillatory tests): η0 as the limiting value of the complex viscosity function η*(ω) “at rest”, i. e. for ω → 0 (see Note 3 in Chapter 8.4.2.1a, Equation 8.27)
3 Creep tests: η0 at the end of the stress phase, when reaching steady-state flow behavior.
6.3.4.2Creep compliance , and creep recovery compliance
Creep (shear) compliance J(t) can be calculated using the following parameters: the preset shear stress τ0 and the resulting deformation function γ(t) obtained in the creep phase. Definition:
Equation 6.5
J(t) = γ(t) / τ0
Unit: [1/Pa = Pa-1].
The (shear) compliance is the reciprocal value of the shear modulus which can be imagined as rigidity then:
Equation 6.6
J = 1/G
The shape of the J(t)-curve is similar to the γ(t)-curve since for creep tests, the shear stress τ0 is preset as a constant value. Usually, J is presented on the y-axis, and time t on the x-axis (see Figure 6.9).
Figure 6.9: Creep compliance curves J(t), a) in the LVE range, b) outside the LVE range, showing comparatively higher J-values then
6.3.4.2.1a) Instantaneous compliance
Definition of the instantaneous (shear) compliance J0 (or steady-state initial creep compliance):
Equation 6.7
J0 = 1 / G0 = γ0 / τ0
Unit: [1/Pa = Pa-1]. The following holds:
Equation 6.8
J0 = limt → 0 J(t)
J0 is the limiting value of the J(t)-function at the very beginning of the creep test, i. e., when t = 0.
G0 [Pa] is the instantaneous (shear) modulus (or steady-state initial shear modulus) representing the sum of the elastic behavior of the springs S1 and S2, and γ0 [%] is the instantaneous deformation, i. e. the limiting value of the deformation at the time point t = 0.
The value of γ0 is determined at the intersection of the γ-axis and the straight line which is fitted to the steady-state sector of the creep curve, showing the curve slope tanβ (see Figure 6.8).
It is the same straight line which is used to determine the value of zero-shear viscosity. To enable accurate determination of the values of J0 (or G0, respectively), at least 10 % of the measuring points of the entire creep curve should occur within this steady-state sector.
Both parameters, J0 and G0, are coefficients to characterize the elastic behavior of a material. Polymers with similar structures but a higher molar mass M are showing lower J-values or higher G-values, respectively. For non-flowing materials, e. g. for chemically strongly crosslinked elastomers such as hard rubbers, the calculated J-values obtained would approach towards a very low value, or the G-value towards a very high value, respectively. Therefore, these kinds of materials would be absolutely rigid, showing hardly yielding.
6.3.4.2.2b) Creep recovery, and equilibrium compliance
Sometimes, the J(t)-function in the rest phase is referred to as the (creep shear) recovery compliance Jr. Definition:
Equation