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Generalized Kelvin/Voigt model
Generalized Kelvin/Voigt models are therefore used to analyze the creep and creep recovery (retardation) functions. A single Kelvin/Voigt model consists of a single spring and a single dashpot in parallel connection (see Figure 5.6). For a generalized Kelvin/Voigt model, however, several Kelvin/Voigt elements are connected in series (see Figure 6.10).
Each one of the individual Kelvin/Voigt elements displays the behavior of an individual polymer fraction having a specific molar mass and molecular structure. Each fraction is represented in the model by a spring and a dashpot which together produce the characteristic values of the viscoelastic behavior of this one fraction. This results in the corresponding individual retardation time Λi.
The following applies to each individual Kelvin/Voigt element:
τi = ηi ⋅ γ ̇ + Gi ⋅ γ
with the individual counting number i = 1 to k; and k is the total number of all Kelvin/Voigt elements available.
The following holds: Λi = ηi/Gi, with the individual retardation time Λi [s]
Thus:
γi(t) = (τ0 / Gi) ⋅ [1 – exp(-t/Λi)]
Dependent on the shear stress step and on its removal, each one of the Kelvin/Voigt elements is showing an individual time-dependent deformation or re-formation behavior, respectively. The resulting total deformation value γ occurs as the sum of all individual deformation values γi:
Equation 6.13
γ(t) = Σi γi(t) = Σi (τ0 / Gi) ⋅ [1 – exp(-t/Λi)]
It is possible to use this calculation since here applies the principle of superposition according to L. Boltzmann (1844 to 1906; see also Chapter 14.2: 1874) [6.7]. According to this principle, for data of the linear viscoelastic range, the ratio of the value pairs of stress/deformation also applies to its multiples and sums (see also Chapter 8.3.2: LVE range) [6.8] [6.9].
Sometimes, an extra spring and an extra dashpot are connected in series as additional components to the generalized Kelvin/Voigt model to enable also analysis, on the one hand of reversible elastic behavior at very low deformations, and on the other hand of purely viscous flow behavior at high deformations (this model is comparable to the Burgers model, see Figure 6.7).
6.3.4.4.2b) Discrete retardation time spectrum
A discrete retardation time spectrum consists of individual value pairs showing a limited total number k of Kelvin/Voigt models connected in series, e. g. with k = 5. In this case, the discrete retardation time spectrum consists of 5 individual value pairs, e. g. expressed in terms of the retardation time-dependent creep compliances Ji (Λi), here with i = 1 to 5. A corresponding example, but using the relaxation time-dependent relaxation moduli, is explained in Chapter 7.3.3.3b, see Table 7.1 and Figure 7.9.
The shape of both exponential functions, the creep and creep recovery curve, is determined by the spectrum of retardation times. For many polymers, the range of retardation times spreads over several decades, and frequently up to much more than 100 s. Often Λi values up to 1000 s (= approx. 17min) or 10,000 s (= approx. 167min = almost 3h) can be found, and sometimes periods which are even longer. Therefore, for samples showing highly viscous or viscoelastic behavior, retardation times of at least half an hour should be taken into account.
Summing up the individual creep compliances Ji(Λi), the function of the time-dependent creep compliance J(t) can be determined in the form of a fitting function:
Equation 6.14
J(t) = Σi (1 / Gi) ⋅ [1 – exp(-t /Λi)]
= (1 / G1) ⋅ [1 – exp(–t /Λ1)] + (1 / G2) ⋅ [1 – exp(–t /Λ2)] + (1 / G3) ⋅ [1 – exp(–t /Λ3)] + ...
The discrete retardation time spectrum can be illustrated in a diagram displaying an individual point for each individual value pair (Λi / Ji). Usually, the Λi values [s] are presented on the x-axis and the Ji values [Pa-1] on the y-axis (similar to Figure 7.9). In the same diagram, the calculated fitting function J(t) may be displayed; here, time t [s] is shown on the x-axis and the J-values [Pa-1] on the y-axis, using the same scale for Λi and t, and also the same scale for Ji and J on the other axis.
6.3.4.4.3c) Continuous retardation time spectrum
The continuous retardation time spectrum H(Λ) is produced from an “infinite” number of individual values i = 1 to k (and k → ∞). H is called the amount or intensity, and H(Λ) is referred to as the distribution function of the retardation times Λ. The sum of the continuous spectrum is usually presented in the form of an integral. The corresponding integral equations are calculated using special analysis programs.
Usually, H(Λ) is presented with H on the y-axis and Λ on the x-axis. The data points of H(Λ) at low or high Λ-values are indicating the number of molecules (or other components) with short or long retardation times, respectively. For information on the relaxation time spectrum H(λ), including a diagram, see Chapter 7.3.3.3c and Figure 7.10; most of the information given there applies also to H(Λ), either directly or in an adapted form.
6.3.5Data conversion
When performing creep tests, data are measurable indeed at such low shear rates which would cause problems in many cases if determined when using other test methods, or the corresponding test would take an extremely long time then. Data of creep tests are often used to characterize long-term behavior. They may be converted, for example from J(t)-values to frequency sweep values at very low frequencies (see Chapter 8.4. for details about frequency sweeps/oscillatory tests).
6.1.2.1.1Example: Conversion of creep test data to frequency sweep data
Data of a creep test are available in terms of a γ(t)-function. The aim is to determine the
corresponding frequency function (oscillatory tests). The following steps are performed:
Using available data of the creep function γ(t)
Calculation of the function of the creep compliance J(t)
Calculation of the continuous retardation time spectrum H(Λ)
Data conversion to determine the corresponding frequency sweep in terms of G’ & G’’(ω)
This method is particularly useful if it is aimed to produce values in the range of zero-shear viscosity (or low-shear range, resp.).
Information given in Chapter 7.3.4 on data conversion and on curve fitting methods also applies here, in the appropriate form, to creep test data. Data conversion can be performed using data of both the time-dependent relaxation modulus G(t) or creep compliance J(t). Relaxation spectrum H(λ) and retardation spectrum H(Λ) correspond to each other approximately. However, it should be taken into account that data which are depending on relaxation times λ or retardation times Λ, respectively, are measured at different shear conditions: for data related to λ with controlled strain (deformation) γ, and for data related to Λ with controlled stress τ. There might be a different response by the structure of the sample when subjected to these different shear conditions.
6.3.6Determination of the molar mass distribution
Using H(Λ) data, special software analysis programs enable users to determine the molar mass distribution (MMD) of uncrosslinked polymers. In a MMD diagram, usually the relative amount
w [%] is presented on the y-axis, and the molar mass M [g/mol] on the x-axis (see Chapter 7.3.5 and Figure 7.11).