8.1.5.1. Arrhenius

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

8.1.5.1. 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.