# Time–temperature superposition

Last updated

The time–temperature superposition principle is a concept in polymer physics and in the physics of glass-forming liquids. [1] [2] [3] [4] This superposition principle is used to determine temperature-dependent mechanical properties of linear viscoelastic materials from known properties at a reference temperature. The elastic moduli of typical amorphous polymers increase with loading rate but decrease when the temperature is increased. [5] Curves of the instantaneous modulus as a function of time do not change shape as the temperature is changed but appear only to shift left or right. This implies that a master curve at a given temperature can be used as the reference to predict curves at various temperatures by applying a shift operation. The time-temperature superposition principle of linear viscoelasticity is based on the above observation. [6]

## Contents

The application of the principle typically involves the following steps:

• experimental determination of frequency-dependent curves of isothermal viscoelastic mechanical properties at several temperatures and for a small range of frequencies
• computation of a translation factor to correlate these properties for the temperature and frequency range
• experimental determination of a master curve showing the effect of frequency for a wide range of frequencies
• application of the translation factor to determine temperature-dependent moduli over the whole range of frequencies in the master curve.

The translation factor is often computed using an empirical relation first established by Malcolm L. Williams, Robert F. Landel and John D. Ferry (also called the Williams-Landel-Ferry or WLF model). An alternative model suggested by Arrhenius is also used. The WLF model is related to macroscopic motion of the bulk material, while the Arrhenius model considers local motion of polymer chains.

Some materials, polymers in particular, show a strong dependence of viscoelastic properties on the temperature at which they are measured. If you plot the elastic modulus of a noncrystallizing crosslinked polymer against the temperature at which you measured it, you will get a curve which can be divided up into distinct regions of physical behavior. At very low temperatures, the polymer will behave like a glass and exhibit a high modulus. As you increase the temperature, the polymer will undergo a transition from a hard “glassy” state to a soft “rubbery” state in which the modulus can be several orders of magnitude lower than it was in the glassy state. The transition from glassy to rubbery behavior is continuous and the transition zone is often referred to as the leathery zone. The onset temperature of the transition zone, moving from glassy to rubbery, is known as the glass transition temperature, or Tg.

In the 1940s Andrews and Tobolsky [7] showed that there was a simple relationship between temperature and time for the mechanical response of a polymer. Modulus measurements are made by stretching or compressing a sample at a prescribed rate of deformation. For polymers, changing the rate of deformation will cause the curve described above to be shifted along the temperature axis. Increasing the rate of deformation will shift the curve to higher temperatures so that the transition from a glassy to a rubbery state will happen at higher temperatures.

It has been shown experimentally that the elastic modulus (E) of a polymer is influenced by the load and the response time. Time–temperature superposition implies that the response time function of the elastic modulus at a certain temperature resembles the shape of the same functions of adjacent temperatures. Curves of E vs. log(response time) at one temperature can be shifted to overlap with adjacent curves, as long as the data sets did not suffer from ageing effects [8] during the test time (see Williams-Landel-Ferry equation).

The Deborah number is closely related to the concept of time-temperature superposition.

## Physical principle

Consider a viscoelastic body that is subjected to dynamic loading. If the excitation frequency is low enough [9] the viscous behavior is paramount and all polymer chains have the time to respond to the applied load within a time period. In contrast, at higher frequencies, the chains do not have the time to fully respond and the resulting artificial viscosity results in an increase in the macroscopic modulus. Moreover, at constant frequency, an increase in temperature results in a reduction of the modulus due to an increase in free volume and chain movement.

Time–temperature superposition is a procedure that has become important in the field of polymers to observe the dependence upon temperature on the change of viscosity of a polymeric fluid. Rheology or viscosity can often be a strong indicator of the molecular structure and molecular mobility. Time–temperature superposition avoids the inefficiency of measuring a polymer's behavior over long periods of time at a specified temperature by utilizing the fact that at higher temperatures and shorter time the polymer will behave the same, provided there are no phase transitions.

## Time-temperature superposition

Consider the relaxation modulus E at two temperatures T and T0 such that T > T0. At constant strain, the stress relaxes faster at the higher temperature. The principle of time-temperature superposition states that the change in temperature from T to T0 is equivalent to multiplying the time scale by a constant factor aT which is only a function of the two temperatures T and T0. In other words,

${\displaystyle E(t,T)=E({\frac {t}{a_{\rm {T}}}},T_{0})\,.}$

The quantity aT is called the horizontal translation factor or the shift factor and has the properties:

{\displaystyle {\begin{aligned}&T>T_{0}\quad \implies \quad a_{\rm {T}}<1\\&T1\\&T=T_{0}\quad \implies \quad a_{\rm {T}}=1\,.\end{aligned}}}

The superposition principle for complex dynamic moduli (G* = G' + i G'') at a fixed frequency ω is obtained similarly:

{\displaystyle {\begin{aligned}G'(\omega ,T)&=G'\left(a_{\rm {T}}\,\omega ,T_{0}\right)\\G''(\omega ,T)&=G''\left(a_{\rm {T}}\,\omega ,T_{0}\right).\end{aligned}}}

A decrease in temperature increases the time characteristics while frequency characteristics decrease.

## Relationship between shift factor and intrinsic viscosities

For a polymer in solution or "molten" state the following relationship can be used to determine the shift factor:

${\displaystyle a_{\rm {T}}={\frac {\eta _{\rm {T}}}{\eta _{\rm {T0}}}}}$

where ηT0 is the viscosity (non-Newtonian) during continuous flow at temperature T0 and ηT is the viscosity at temperature T.

The time–temperature shift factor can also be described in terms of the activation energy (Ea). By plotting the shift factor aT versus the reciprocal of temperature (in K), the slope of the curve can be interpreted as Ea/k, where k is the Boltzmann constant = 8.64x10−5 eV/K and the activation energy is expressed in terms of eV.

## Shift factor using the Williams-Landel-Ferry (WLF) model

The empirical relationship of Williams-Landel-Ferry, [11] combined with the principle of time-temperature superposition, can account for variations in the intrinsic viscosity η0 of amorphous polymers as a function of temperature, for temperatures near the glass transition temperature Tg. The WLF model also expresses the change with the temperature of the shift factor.

Williams, Landel and Ferry proposed the following relationship for aT in terms of (T-T0) :

${\displaystyle \log a_{\rm {T}}=-{\frac {C_{1}(T-T_{0})}{C_{2}+(T-T_{0})}}}$

where ${\displaystyle \log }$ is the decadic logarithm and C1 and C2 are positive constants that depend on the material and the reference temperature. This relationship holds only in the approximate temperature range [Tg, Tg + 100 °C]. To determine the constants, the factor aT is calculated for each component M′ and M of the complex measured modulus M*. A good correlation between the two shift factors gives the values of the coefficients C1 and C2 that characterize the material.

If T0 = Tg:

${\displaystyle \log a_{\rm {T}}=-{\frac {C_{1}^{g}(T-T_{g})}{C_{2}^{g}+(T-T_{g})}}=\log \left({\frac {\eta _{\rm {T}}}{\eta _{T_{g}}}}\right)}$

where Cg1 and Cg2 are the coefficients of the WLF model when the reference temperature is the glass transition temperature.

The coefficients C1 and C2 depend on the reference temperature. If the reference temperature is changed from T0 to T′0, the new coefficients are given by

${\displaystyle C'_{1}={\frac {C_{1}\,C_{2}}{C_{2}+(T'_{0}-T_{0})}}\qquad {\rm {and}}\qquad C'_{2}=C_{2}+(T'_{0}-T_{0})\,.}$

In particular, to transform the constants from those obtained at the glass transition temperature to a reference temperature T0,

${\displaystyle C_{1}^{0}={\frac {C_{1}^{g}\,C_{2}^{g}}{C_{2}^{g}+(T_{0}-T_{g})}}\qquad {\rm {and}}\qquad C_{2}^{0}=C_{2}^{g}+(T_{0}-T_{g})\,.}$

These same authors have proposed the "universal constants" Cg1 and Cg2 for a given polymer system be collected in a table. These constants are approximately the same for a large number of polymers and can be written Cg1 ≈ 15 and Cg2 ≈ 50 K. Experimentally observed values deviate from the values in the table. These orders of magnitude are useful and are a good indicator of the quality of a relationship that has been computed from experimental data.

## Construction of master curves

The principle of time-temperature superposition requires the assumption of thermorheologically simple behavior (all curves have the same characteristic time variation law with temperature). From an initial spectral window [ω1, ω2] and a series of isotherms in this window, we can calculate the master curves of a material which extends over a broader frequency range. An arbitrary temperature T0 is taken as a reference for setting the frequency scale (the curve at that temperature undergoes no shift).

In the frequency range [ω1, ω2], if the temperature increases from T0, the complex modulus E′(ω) decreases. This amounts to explore a part of the master curve corresponding to frequencies lower than ω1 while maintaining the temperature at T0. Conversely, lowering the temperature corresponds to the exploration of the part of the curve corresponding to high frequencies. For a reference temperature T0, shifts of the modulus curves have the amplitude log(aT). In the area of glass transition, aT is described by an homographic function of the temperature.

The viscoelastic behavior is well modeled and allows extrapolation beyond the field of experimental frequencies which typically ranges from 0.01 to 100 Hz .

## Shift factor using Arrhenius law

The shift factor (which depends on the nature of the transition) can be defined, below Tg,[ citation needed ] using an Arrhenius law:

${\displaystyle \log(a_{\rm {T}})=-{\frac {E_{a}}{2.303R}}\left({\frac {1}{T}}-{\frac {1}{T_{0}}}\right)}$

where Ea is the activation energy, R is the universal gas constant, and T0 is a reference temperature in kelvins. This Arrhenius law, under this glass transition temperature, applies to secondary transitions (relaxation) called β-transitions.

## Limitations

For the superposition principle to apply, the sample must be homogeneous, isotropic and amorphous. The material must be linear viscoelastic under the deformations of interest, i.e., the deformation must be expressed as a linear function of the stress by applying very small strains, e.g. 0.01%.

To apply the WLF relationship, such a sample should be sought in the approximate temperature range [Tg, Tg + 100 °C], where α-transitions are observed (relaxation). The study to determine aT and the coefficients C1 and C2 requires extensive dynamic testing at a number of scanning frequencies and temperature, which represents at least a hundred measurement points.

## Related Research Articles

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

A viscometer is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be considered as a special type of viscometer. Viscometers can measure only constant viscosity, that is, viscosity that does not change with flow conditions.

The Deborah number (De) is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It quantifies the observation that given enough time even a solid-like material might flow, or a fluid-like material can act solid when it is deformed rapidly enough. Materials that have low relaxation times flow easily and as such show relatively rapid stress decay.

In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response, and a Bode phase plot, expressing the phase shift.

Dynamic mechanical analysis is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the complex modulus. The temperature of the sample or the frequency of the stress are often varied, leading to variations in the complex modulus; this approach can be used to locate the glass transition temperature of the material, as well as to identify transitions corresponding to other molecular motions.

Hemorheology, also spelled haemorheology, or blood rheology, is the study of flow properties of blood and its elements of plasma and cells. Proper tissue perfusion can occur only when blood's rheological properties are within certain levels. Alterations of these properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit and mechanical properties of red blood cells. Red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. Because of that, blood behaves as a non-Newtonian fluid. As such, the viscosity of blood varies with shear rate. Blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. Therefore, blood is a shear-thinning fluid. Contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. Blood viscosity also increases with increases in red cell aggregability.

In materials science, creep is the tendency of a solid material to undergo slow deformation while subject to persistent mechanical stresses. It can occur as a result of long-term exposure to high levels of stress that are still below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods and generally increase as they near their melting point.

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

A quartz crystal microbalance (QCM) measures a mass variation per unit area by measuring the change in frequency of a quartz crystal resonator. The resonance is disturbed by the addition or removal of a small mass due to oxide growth/decay or film deposition at the surface of the acoustic resonator. The QCM can be used under vacuum, in gas phase and more recently in liquid environments. It is useful for monitoring the rate of deposition in thin film deposition systems under vacuum. In liquid, it is highly effective at determining the affinity of molecules to surfaces functionalized with recognition sites. Larger entities such as viruses or polymers are investigated as well. QCM has also been used to investigate interactions between biomolecules. Frequency measurements are easily made to high precision ; hence, it is easy to measure mass densities down to a level of below 1 μg/cm2. In addition to measuring the frequency, the dissipation factor is often measured to help analysis. The dissipation factor is the inverse quality factor of the resonance, Q−1 = w/fr ; it quantifies the damping in the system and is related to the sample's viscoelastic properties.

Dynamic modulus is the ratio of stress to strain under vibratory conditions. It is a property of viscoelastic materials.

Thermomechanical analysis (TMA) is a technique used in thermal analysis, a branch of materials science which studies the properties of materials as they change with temperature.

The McCumber relation is a relationship between the effective cross-sections of absorption and emission of light in the physics of solid-state lasers. It is named after Dean McCumber, who proposed the relationship in 1964.

An acoustic rheometer employs a piezo-electric crystal that can easily launch a successive wave of extensions and contractions into the fluid. It applies an oscillating extensional stress to the system. System response can be interpreted in terms of extensional rheology.

In polymer chemistry the kinetic chain length of a polymer, ν, is the average number of units called monomers added to a growing chain during chain-growth polymerization. During this process, a polymer chain is formed when monomers are bonded together to form long chains known as polymers. Kinetic chain length is defined as the average number of monomers that react with an active center such as a radical from initiation to termination.

The glass–liquid transition, or glass transition, is the gradual and reversible transition in amorphous materials from a hard and relatively brittle "glassy" state into a viscous or rubbery state as the temperature is increased. An amorphous solid that exhibits a glass transition is called a glass. The reverse transition, achieved by supercooling a viscous liquid into the glass state, is called vitrification.

Microrheology is a technique used to measure the rheological properties of a medium, such as microviscosity, via the measurement of the trajectory of a flow tracer. It is a new way of doing rheology, traditionally done using a rheometer. There are two types of microrheology: passive microrheology and active microrheology. Passive microrheology uses inherent thermal energy to move the tracers, whereas active microrheology uses externally applied forces, such as from a magnetic field or an optical tweezer, to do so. Microrheology can be further differentiated into 1- and 2-particle methods.

The Williams–Landel–Ferry Equation is an empirical equation associated with time–temperature superposition.

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

In polymer chemistry and polymer physics, the Flory–Fox equation is a simple empirical formula that relates molecular weight to the glass transition temperature of a polymer system. The equation was first proposed in 1950 by Paul J. Flory and Thomas G. Fox while at Cornell University. Their work on the subject overturned the previously held theory that the glass transition temperature was the temperature at which viscosity reached a maximum. Instead, they demonstrated that the glass transition temperature is the temperature at which the free space available for molecular motions achieved a minimum value. While its accuracy is usually limited to samples of narrow range molecular weight distributions, it serves as a good starting point for more complex structure-property relationships.

Rheological weldability (RW) of thermoplastics considers the materials flow characteristics in determining the weldability of the given material. The process of welding thermal plastics requires three general steps, first is surface preparation. The second step is the application of heat and pressure to create intimate contact between the components being joined and initiate inter-molecular diffusion across the joint and the third step is cooling. RW can be used to determine the effectiveness of the second step of the process for given materials.

## References

1. Hiemenz, Paul C.; Lodge, Timothy P. (2007). Polymer chemistry (2nd ed.). Taylor & Francis. pp. 486–491. ISBN   1574447793.
2. Li, Rongzhi (February 2000). "Time-temperature superposition method for glass transition temperature of plastic materials". Materials Science and Engineering: A. 278 (1–2): 36–45. doi:10.1016/S0921-5093(99)00602-4.
3. van Gurp, Marnix; Palmen, Jo (January 1998). "Time-temperature superposition for polymeric blends" (PDF). Rheology Bulletin. 67 (1): 5–8. Retrieved 7 December 2021.
4. Olsen, Niels Boye; Christensen, Tage; Dyre, Jeppe C. (2001). "Time-Temperature Superposition in Viscous Liquids". Physical Review Letters. 86 (7): 1271–1274. arXiv:. Bibcode:2001PhRvL..86.1271O. doi:. ISSN   0031-9007. PMID   11178061.
5. Experiments that determine the mechanical properties of polymers often use periodic loading. For such situations, the loading rate is related to the frequency of the applied load.
6. Christensen, Richard M. (1971). Theory of viscoelasticity: an introduction. New York: Academic Press. p. 92. ISBN   0121742504.
7. Andrews, R. D.; Tobolsky, A. V. (August 1951). "Elastoviscous properties of polyisobutylene. IV. Relaxation time spectrum and calculation of bulk viscosity". Journal of Polymer Science. 7 (23): 221–242. doi:10.1002/pol.1951.120070210.
8. Struik, L. C. E. (1978). Physical aging in amorphous polymers and other materials. Amsterdam: Elsevier Scientific Pub. Co. ISBN   9780444416551.
9. For the superposition principle to apply, the excitation frequency should be well above the characteristic time τ (also called relaxation time) which depends on the molecular weight of the polymer.
10. Curve has been generated with data from a dynamic test with a double scanning frequency / temperature on a viscoelastic polymer.
11. Ferry, John D. (1980). Viscoelastic properties of polymers (3d ed.). New York: Wiley. ISBN   0471048941.