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

## Arrhenius

The Arrhenius model predicts failure acceleration due to temperature increase

One of the earliest and most successful acceleration models predicts how time-to-fail varies with temperature. This empirically based model is known as the Arrhenius equation. It takes the form  $$t_f = A \cdot \mbox{exp} \left[ \frac{\Delta H}{k T} \right] \,\, ,$$ with $$T$$ denoting temperature measured in degrees Kelvin (273.16 + degrees Celsius) at the point when the failure process takes place and $$k$$ is Boltzmann's constant (8.617e-5 in ev/K). The constant $$A$$ is a scaling factor that drops out when calculating acceleration factors, with $$\Delta H$$ (pronounced "Delta H") denoting the activation energy, which is the critical parameter in the model.

The Arrhenius activation energy, $$\Delta H$$, is all you need to know to calculate temperature acceleration The value of $$\Delta H$$ depends on the failure mechanism and the materials involved, and typically ranges from 0.3 or 0.4 up to 1.5, or even higher. Acceleration factors between two temperatures increase exponentially as $$\Delta H$$ increases.

The acceleration factor between a higher temperature $$T_2$$ and a lower temperature $$T_1$$ is given by $$AF = \mbox{exp} \left[ \frac{\Delta H}{k} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \right] \,\, .$$

Using the value of $$k$$ given above, this can be written in terms of $$T$$ in degrees Celsius as  $$AF = \mbox{exp} \left[ \Delta H \cdot 11605 \cdot \left( \frac{1}{T_1 + 273.16} - \frac{1}{T_2 + 273.16} \right) \right] \,\, .$$

Note that the only unknown parameter in this formula is $$\Delta H$$.

Example: The acceleration factor between 25°C and 125°C is 133 if $$\Delta H$$ = 0.5 and 17,597 if $$\Delta H$$ = 1.0.

The Arrhenius model has been used successfully for failure mechanisms that depend on chemical reactions, diffusion processes or migration processes. This covers many of the non-mechanical (or non-material fatigue) failure modes that cause electronic equipment failure.