8. Assessing Product Reliability
8.1. Introduction
8.1.5. What are some common acceleration models?

## Eyring

The Eyring model has a theoretical basis in chemistry and quantum mechanics and can be used to model acceleration when many stresses are involved Henry Eyring's contributions to chemical reaction rate theory have led to a very general and powerful model for acceleration known as the Eyring Model. This model has several key features:
• It has a theoretical basis from chemistry and quantum mechanics.
• If a chemical process (chemical reaction, diffusion, corrosion, migration, etc.) is causing degradation leading to failure, the Eyring model describes how the rate of  degradation varies with stress or, equivalently, how time to failure varies with stress.
• The model includes temperature and can be expanded to include other relevant stresses.
• The temperature term by itself is very similar to the Arrhenius empirical model, explaining why that model has been so successful in establishing the connection between the $$\Delta H$$ parameter and the quantum theory concept of "activation energy needed to cross an energy barrier and initiate a reaction".
The model for temperature and one additional stress takes the general form: $$t_f = A T^\alpha \mbox{exp} \left[ \frac{\Delta H}{k T} + \left(B + \frac{C}{T} \right) \cdot S_1 \right] \,\, ,$$ for which $$S_1$$ could be some function of voltage or current or any other relevant stress and the parameters $$\alpha, \Delta H, B,$$ and $$C$$ determine acceleration between stress combinations. As with the Arrhenius Model, $$k$$ is Boltzmann's constant and temperature is in degrees Kelvin.

If we want to add an additional non-thermal stress term, the model becomes $$t_f = A T^\alpha \mbox{exp} \left[ \frac{\Delta H}{kT} + \left( B + \frac{C}{T} \right) \cdot S_1 + \left(D + \frac{E}{T} \right) \cdot S_2 \right] \,\, ,$$

and as many stresses as are relevant can be included by adding similar terms.

Models with multiple stresses generally have no interaction terms - which means you can multiply acceleration factors due to different stresses Note that the general Eyring model includes terms that have stress and temperature interactions (in other words, the effect of changing temperature varies, depending on the levels of other stresses). Most models in actual use do not include any interaction terms, so that the relative change in acceleration factors when only one stress changes does not depend on the level of the other stresses.

In models with no interaction, you can compute acceleration factors for each stress and multiply them together. This would not be true if the physical mechanism required interaction terms - but, at least to first approximations, it seems to work for most examples in the literature.

The Eyring model can also be used to model rate of degradation leading to failure as a function of stress Advantages of the Eyring Model
• Can handle many stresses.
• Can be used to model degradation data as well as failure data.
• The $$\Delta H$$ parameter has a physical meaning and has been studied and estimated for many well known failure mechanisms and materials.
In practice, the Eyring Model is usually too complicated to use in its most general form and must be "customized" or simplified for any particular failure mechanism Disadvantages of the Eyring Model
• Even with just two stresses, there are 5 parameters to estimate. Each additional stress adds 2 more unknown parameters.
• Many of the parameters may have only a second-order effect. For example, setting $$\alpha$$ = 0 works quite well since the temperature term then becomes the same as in the Arrhenius model. Also, the constants $$C$$ and $$E$$ are only needed if there is a significant temperature interaction effect with respect to the other stresses.
• The form in which the other stresses appear is not specified by the general model and may vary according to the particular failure mechanism. In other words, $$S_1$$ may be voltage or ln (voltage) or some other function of voltage.
Many well-known models are simplified versions of the Eyring model with appropriate functions of relevant stresses chosen for $$S_1$$ and $$S_2$$. Some of these will be shown in the Other Models section. The trick is to find the right simplification to use for a particular failure mechanism.