Draft:Eyring equation for polymers

• Comment: Viscous flow and stress relaxation sections need citations. You can learn more at WP:Cite Shocksingularity (talk) 21:43, 9 September 2025 (UTC)

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In 1935, Henry Eyring developed a model to describe the rate of thermally activated processes, currently known as the Eyring equation. The theory was initially used for calculations on chemical reactions but was later also applied to other processes in which an energy barrier has to be passed. The original equation is as follows:

$k={\frac {k_{B}T}{h}}e^{-\Delta G^{\ddagger }/RT}$

Where:

• k = rate constant
• k_B = Boltzmann constant
• T = absolute temperature
• h = Planck's constant
• $\Delta G^{\ddagger }$ = Gibb's free energy
• R = gas constant

In 1936 Eyring modified the equation in order to describe the viscoelastic behaviour of polymers:.. This is because, also with polymers, energy barriers must be overcome for deformation or displacement (flow). The theory describes the jump of segments of macromolecules over an energy barrier, causing either deformation in the glass and rubber phase or plastic flow in the melt phase. The resulting Eyring flow theory yields the following equation:

${\dot {\epsilon }}={\dot {\epsilon }}_{0}\exp \left(-{\frac {\Delta E_{a}}{k_{B}T}}\right)\sinh \left({\frac {\sigma V_{a}}{k_{B}T}}\right)$

Where:

• ${\dot {\epsilon }}$ = strain rate or shear rate
• ${\dot {\epsilon }}_{0}$ = pre-exponential factor related to the frequency of the molecular movement
• $\Delta {E_{a}}$ = activation energy needed to start the molecular movement
• k_B = Boltzmann constant
• T = absolute temperature
• σ = tensile stress or shear stress
• V_a = activation volume

The activation volume is related to the volume of the chain segments that are moved into other positions while passing the energy barrier. The product of activation volume with stress equals the energy that is released during a jump into another position. It effectively reduces the energy that is needed to pass the barrier for the jump (E_eff = ΔE_a - σV_a). In glassy polymers the activation volumes are in the order of several cubic nanometers

Application for polymers

The Eyring flow equation offers a molecular model that describes the behaviour of polymers on a macroscopic scale. For the glass and the rubber phase it describes the rotations of chain segments during deformation. In the melt phase it describes the reptation steps of the macromolecules during flow. Some typical applications are:

• Yield stress (glass phase).
• Viscous flow (melt phase).
• Stress relaxation (glass, rubber and melt phase).

Yield stress of a polymer in the glass phase

In the glass phase the polymer molecules are stiff and immobile. Yet, parts of the molecules can move or rotate a little bit in order to accommodate to the circumstances. If the polymer is subjected to a stress that is high enough then chain segments will start to rotate (align) in the direction of the stress and the polymer molecules will deform. This stress is called the yield stress.

The standard Eyring equation above can be rewritten for describing the yield stress behaviour

$\sigma ={\frac {{k_{B}}T}{V_{a}}}\sinh ^{-1}{\biggl [}{\frac {\dot {\epsilon }}{{\dot {\epsilon }}_{0}}}\exp {\biggl (}{\frac {\Delta E_{a}}{k_{B}T}}{\biggr )}{\biggr ]}$

In this equation σ now represents the yield stress, ΔE_a denotes the energy barrier that has to be passed during a conformational change of a chain segment and V_a is related to the volume that is moved during that conformational change.

Fig. 1: Graphical representation of the yield stress (linear) of a polymer versus the shear rate (logarithmic).

For large values of the argument the inverse hyperbolic sine function becomes almost equal to a logarithm:

$\sinh ^{-1}(x)=\ln {\bigl (}x+{\sqrt {1+x^{2}}}{\bigr )}\approx \ln(2x)$ when x >> 1

That means that for large strain rates the equation for the yield stress can be simplified to:

$\sigma \thickapprox {\frac {{k_{B}}T}{V_{a}}}\ln {\Bigl (}2{\frac {\dot {\epsilon }}{{\dot {\epsilon }}_{0}}}{\Bigr )}+{\frac {\Delta E_{a}}{V_{a}}}$

This equation shows that at higher deformation speeds the yield stress increases logarithmically with the strain rate as illustrated in figure 1 above. This logarithmic behaviour is often observed experimentally.

Viscous flow of a polymer in the melt phase

In the melt phase the polymer molecules are very mobile. The thermal energy is by far high enough to make the chain segments rotate at very high speeds. This makes the macromolecules flexible, like a necklace. Due to all the rotating chain segments the macromolecules are able to shift randomly along their axis into other positions like reptiles. This kind of movement of the macromolecules is called reptation.

Due to this reptation the macromolecules are free to move. The material has become a fluid with a certain viscosity. According to Newton's law of viscosity the viscosity is defined as the ratio of stress and shear rate (η = σ/ἑ). Now we can modify the Eyring relation into:

$\eta ={\frac {1}{{\dot {\epsilon }}_{0}}}\exp {\Bigl (}{\frac {\Delta E_{a}}{k_{B}T}}{\Bigr )}{\frac {\sigma }{\sinh {{\Bigl (}{\frac {\sigma V_{a}}{k_{B}T}}{\Bigr )}}}}$

In this equation η now represents the viscosity, ΔE_a is the energy barrier that has to be passed during reptation of the macromolecule into another position and V_a is related to the volume of the reptating macromolecule and σ is the shear or tensile stress applied to the melt.

Fig. 2: Graphical representation of the viscosity (logarithmic) of a polymer versus the shear stress (linear).

According to the equation above the viscosity is at first constant when the stress is low.

For higher stresses the viscosity reduces exponentially with the applied stress. The stress actually facilitates the reptation of the macromolecules because it reduces the energy barrier for reptation. A graphical representation of the viscosity versus the stress is shown in figure 2.

Fig. 3: Graphical representation of the viscosity (logarithmic) of a polymer versus the shear rate (logarithmic).

It is quit common however to measure the viscosity of the molten polymer in relation to the shear rate and not the stress. By substituting $\sigma =\eta {\dot {\epsilon }}$ into the equation above we find after some rewriting:

$\eta ={\frac {k_{B}T}{V_{a}{\dot {\epsilon }}}}\sinh ^{-1}{\Bigl [}{\frac {\dot {\epsilon }}{\dot {\epsilon _{0}}}}\exp {\Bigl (}{\frac {\Delta {E_{a}}}{k_{B}T}}{\Bigr )}{\Bigr ]}$

At very low shear rates we find for the zero shear viscosity:

$\eta ={\frac {k_{B}T}{V_{a}{\dot {\epsilon }}_{0}}}\exp {\Bigl (}{\frac {\Delta E_{a}}{k_{B}T}}{\Bigr )}$

A graphical representation of the viscosity versus the shear rate according to this equation is shown in figure 3. With increasing shear rates the viscosity is at first constant and then starts to reduce. In the limit the viscosity is inversely proportional with the shear rate.

Stress relaxation

According to the Maxwell model and the Generalized Maxwell model the relaxation time (τ) can be calculated as the ratio of viscosity (η) and elasticity modulus (E):

$\tau ={\frac {\eta }{E}}$

Together with the relation for the viscosity above we then obtain for the relaxation time:

$\tau ={\frac {1}{E{\dot {\epsilon }}_{0}}}\exp {\Bigl (}{\frac {\Delta E_{a}}{k_{B}T}}{\Bigr )}{\frac {\sigma }{\sinh {{\Bigl (}{\frac {\sigma V_{a}}{k_{B}T}}{\Bigr )}}}}$

This equation shows that the relaxation time is strongly dependant on both the temperature (T) and the applied stress (σ). Actually the relaxation time reduces exponentially with the stress. Such as relation has been discussed extensively by J.J.M. Baltussen and M.G. Northolt in their publication "The Eyring reduced time model for viscoelastic and yield deformation of polymer fibres". They also suggested a simple differential equation for calculating the time dependence of the stress in a deformed body:

${\frac {d\sigma }{dt}}=-{\frac {\sigma }{\tau }}$

Fig. 4: Stress (σ) reduces with time (t) in a deformed plastic body. The relaxation is fast in the beginning and slows down afterwards.

This differential equation, together with the relation for the relaxation time (τ), gives a result as shown in figure 4. Relaxation of stress in a deformed body will be relatively fast in the beginning while the stress is high and will slow down later when the stress reduces.

All processes that are related to either deformation or flow of polymer molecules will speed up when stress is applied. Typical examples of such processes are:

• Creep of polymer parts under stress
• Yielding of polymer parts under stress
• Viscosity reduction at high shear rates