Accelerated testing is crucial in the development and quality control of many devices including batteries, integrated circuits, microprocessors, etc. Anything that has a multi-year shelf-life or life cycle.
Alkaline batteries, for example, are often advertized to have a 10-year shelf-life. The only way to test such an extreme shelf-life and to verify this claim is through accelerated test methods. To be sure, real time testing is also done as a final verification, but by the time those results are in, the batteries have long since left the manufacturing plant and are in the hands of customers. The last thing any manufacturer wants is for customers to find the flaws in their products. Thus, manufacturers must have test methods that root out flaws and defects long before customers ever see them.
The most often used accelerator in accelerated testing methods is temperature. By increasing the temperature, degradation reactions are sped up and failure is accelerated. By testing to failure at multiple temperatures, the time-to-failure can be extrapolated back to room temperature where real-time testing is impractical for quality control or product development activities.
Here is the scientific foundation for elevated temperature accelerated testing.
The Arrhenius Theory:
Accelerated testing using elevated temperature is based on the theory first proposed in 1884 by the Dutch chemist J.H. van’t Hoff and later updated with a physical interpretation in 1889 by Svante Arrhenius, a Swedish Physicist and chemist. Arrhenius proposed that molecules must be in an activated state to react and that temperature enhances the rate of a reaction by increasing the fraction of molecules in the activated state.
The energy needed to raise one mol of molecules to the activated state is called the Activation Energy and the equation relating the rate constant of a reaction to the activation energy is as follows, where the rate constant is a temperature-dependent proportionality factor that defines how the rate of the reaction will vary with temperature:
kT = Ae-Ea/RT [1]
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kT = Ae-Ea/RT [1]
kT = Ae-Ea/RT | [1] |
———————–
kT = Ae-Ea/RT [1]
kT = Ae-Ea/RT [1]
kT = Ae-Ea/RT [1]
kT = Ae-Ea/RT [1]
kT = Ae-Ea/RT [1]
XXXXXXXXXXXXXXXXXXXXXXXXXX
Where
kT = The rate constant of the reaction at temperature (T) (sec-1)
A = Pre-exponential factor (sec-1) (often referred to as the frequency factor since it measures the effective frequency of molecular collisions)
e = Base of natural logarithms (2.71828)
Ea = Activation Energy (J/mol)
R = Gas constant (8.3143 J / mol-oK ) (note: if the activation energy is being expressed for individual molecules, then the Boltzmann constant would be used, which is 1.3806 x 10-23 J/oK)
T = Temperature (oK)
For a chemical reaction, such as:
aA + bB ———> cC + dD [2]
The rate of the reaction at temperature (T) is given by:
vT = kT[A]a [B]b [3]
where
vT = reaction rate at temperature (T) (mol/sec)
kT = Rate constant of reaction at temperature (T) (sec-1)
[A] = Concentration of reactant A (mol/l)
[B] = Concentration of reactant B (mol/l)
a = Stoichiometric coefficient for Reactant A
b = Stoichiometric coefficient for Reactant B
Assuming the mechanism is the same at all temperatures evaluated, the concentration terms in Equation [3] can be represented by a constant, giving:
vT = kT[A]a[B]b = kTC ……………………………………………………………………………………….. [4]
where
C = [A]a[B]b
Substituting for kT from Equation [1] and combining constants gives:
vT = kTC = C(Ae-Ea/RT ) = CAe-Ea/RT = C’e-Ea/RT [5]
Where
vT = Reaction rate at temperature (T) (mol/sec)
kT = Rate constant at temperature (T) (sec-1)
C = [A]a[B]b (concentration term)
C’ = CA (modified pre-exponential factor containing the concentration term)
Equation [5] is commonly referred to as the Arrhenius equation and, in logarithmic form, becomes:
Ln(vT) = Ln(C’) – Ea/RT [6]
Determining Ea and C’ From Test Data:
In accelerated testing, the exact reaction mechanism is often not known, so one empirically measures the reaction rate (or time to failure) at several different temperatures and then analyzes the resulting data by plotting Ln(vT) versus 1/T.
For time-to-failure measurements, the reaction rate would be given by the reciprocal of the time to failure.
According to Equation [6], the resulting graph should yield a straight line having a slope of –Ea/R and an intercept of Ln(C’), thereby, allowing the activation Energy (Ea) and the modified pre-exponential constant (C’) to be to be readily obtained.
This plot also allows verification that a single mechanism operates over the range of temperatures evaluated. A single mechanism is a crucial requirement for temperature-based accelerated testing and any significant deviation from linearity or, more significantly, an inflection point, would indicate multiple mechanisms and would invalidate conclusions based on data obtained at temperatures above the linear part of the graph or above the inflection point. This is illustrated in Figure 1.
Calculating Reaction Rates at Lower Temperatures:
Once the activation energy and modified pre-exponential factors have been obtained, the reaction rate (or time to failure) can be calculated for other temperatures of interest, particularly at lower temperatures where the rates are too slow and take too long to measure empirically.
A Useful Approximation:
For chemical reactions, the general rule of thumb is that the reaction rate will double for each 10oC rise in temperature. In this case, the time to failure (TTF) at some temperature (T2) can be estimated from the time to failure measured at an elevated test temperature (T1) using the following equation.
TTF T2 = TTF T1 * 2((T1-T2)/10) [7]
Figure 2 shows a plot of time to failure versus temperature for this equation. This relationship is often used in the battery industry and shows that 28 days at 71C is equivalent to two years at ambient temperature (24C).
Wrapping Up:
Accelerated test methods are an important part of our technology development in the modern era.
Although the above rule of thumb is a useful guideline, whenever possible, it is best to actually measure the reaction rates or times-to-failure at several different temperatures and go through the Arrhenius analysis of the data.
In addition to chemical reactions, many physical processes have activation energies associated with them and can be modeled using the Arrhenius Equation. These include diffusion, fluid viscosity, solution conductivity, and semiconductor conductivity.