The Role of Differential Calculus in Understanding Oxidation and Material Stability

When a piece of metal is exposed to air, it immediately begins to react with oxygen. This reaction forms a thin oxide layer that helps protect the metal from further corrosion. These oxides are crucial in industries such as aerospace, energy, and manufacturing, where materials must withstand long-term exposure to heat, moisture, and chemicals. Among them, titanium and steel are often studied for how their oxide layers form and evolve, providing insight into the broader field of corrosion prevention.

Process and Mathematical Model

The growth of an oxide film is not constant over time. Initially, the layer thickens rapidly, but as it becomes denser, the rate slows. This slowing effect occurs because oxygen ions must diffuse through the existing oxide to reach the underlying metal. The relationship between time and film thickness can be modelled mathematically using a differential equation that describes how the growth rate decreases as the film thickens. This equation not only explains why metals naturally develop protective coatings but also helps engineers predict material lifetimes in harsh environments.

We start with the assumption that the rate of growth is inversely proportional to thickness:

where x(t) is the oxide thickness at time t, and k is a constant that captures how easily atoms can diffuse through the oxide. The constant k is a composite function, determined by the ambient temperature and alloy composition. By separating the variables and integrating, we can form a function for the oxide thickness.

In this instance, the constant of integration is interpreted as the initial oxide thickness (x₀).

This is known as the parabolic law of oxidation: thickness grows with the square root of time.

The ordinary differential equation explains the everyday observation that rusting slows down. At first, there are greater moles of metal available for oxygen to react with, so growth is fast. As the barrier thickens, diffusion through it limits the rate, and the curve of x versus t bends gently toward a plateau.

Figure 1: Representing Oxide thickness as a function of time, at different temperatures.

This law can be applied to any given material. Let’s take steel as an example to find the oxidation layer thickness growth per year in a tropical climate such as Bangkok. Assume T= 28°C and the abundance of oxygen in the atmosphere is negligible and constant. We’ll also assume that x₀is negligible because newly produced steel wouldn’t have enough exposure to oxygen to form a preexisting coat.

k can be calculated as follows, using the Arrhenius function with reference to the parameters mentioned earlier.

k₀ = the pre exponential term, which represents the maximum possible rate constant if there were no energy barrier

e = Euler’s number

Ea = Activation energy

R = Ideal Gas constant

T = Temperature, in Kelvin

For steel oxidation:

By substituting into the Arrhenius equation:

Now that the value for k has been found, we can substitute k back into the parabolic rate equation for t = 3.15 x 10⁷:

The oxidation layer formed is only 2.5 nanometres per year, which explains why steel is such a versatile material for large structures.

The mathematics behind oxide growth transforms a surface reaction into something quantifiable. By expressing corrosion as an ordinary differential equation, we reveal how the interplay between time, temperature, and diffusion dictates a material’s stability. The parabolic law is more than a formula. It's a predictive tool that turns microscopic movement of atoms into measurable engineering outcomes.

For materials scientists, this relationship is invaluable. The rate constant k encapsulates key physical parameters: diffusion coefficients, activation energy, and alloy composition. By analysing how k evolves with temperature through the Arrhenius function, we can forecast how steels, oxides, or even semiconductors will behave under different environmental conditions.

In essence, differential calculus provides a rigorous framework that unifies theoretical understanding with practical application in materials science. Through the process of integration, the oxidation of steel can be expressed as a quantifiable relationship between time, temperature, and atomic diffusion. This mathematical formalism transforms corrosion from an empirical observation into a predictable and optimisable phenomenon. By employing calculus, materials scientists can derive rate laws, estimate diffusion coefficients, and forecast structural stability. This demonstrates that the durability of a material is not merely a chemical property, but a consequence of well-defined mathematical principles. 

Bibliography

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Climate.Top (2024). Bangkok Climate Bangkok Temperatures Bangkok, Thailand Weather Averages. [online] Climate.top. Available at: https://www.climate.top/thailand/bangkok/ [Accessed 4 Oct. 2025].