The Heat Equation: The Theoretical Basis for Materials Processing

Dear Colleagues,

Two centuries have passed since Fourier introduced the heat equation, yet the subject continues to be of significant scientific interest. This sustained relevance is driven by ongoing advancements in both the mathematical theory of heat conduction and the expanding technological domains that rely on its applications. The heat equation can be studied through a variety of modeling approaches, including analytical, semi-analytical, semi-analytical-numerical, and fully numerical methods. Among numerical tools, the finite element method stands out due to its versatility and accuracy, and it is effectively implemented using advanced software packages such as COMSOL Multiphysics.

Light has long held a central role across the disciplines of physics, chemistry, and biology. Over the last century, the advent of lasers—devices that stimulate atoms and molecules to emit and amplify light at specific wavelengths—has driven transformative advances in fields such as medicine, industrial material processing, data storage, printing, and defense. In each of these applications, the interaction between laser light and solid matter is of fundamental importance. To model and understand these interactions, the classical heat equation has been widely employed, serving as a foundational tool in numerous theoretical and practical analyses.

Despite certain criticisms regarding its limitations in extreme or ultrafast regimes, the classical heat equation remains a powerful and widely used tool for describing thermal effects in laser‒solid interactions. It is applicable to both homogeneous and inhomogeneous materials, providing valuable insights into heat transport dynamics. Particular attention has been devoted to its application in multi-layered structures and thin films, where thermal gradients and interfacial effects play a critical role in governing the overall response to laser irradiation.

Over the years, various Fourier and non-Fourier heat conduction models—incorporating thermal relaxation times—have been developed to address the limitations of classical approaches, particularly under conditions of rapid energy deposition. Among these, the two-temperature model (TTM), introduced nearly five decades ago, has become a cornerstone in the study of laser‒solid interactions. The TTM accounts for the distinct thermal responses of the electron and lattice subsystems, making it especially relevant for modeling ultrafast laser irradiation of metals. This model has significantly influenced the field and continues to be the subject of active investigation. Ultra-short laser processing has emerged as a widely explored researched topic due to its broad range of applications, spanning industry, medicine, fundamental research, and defense. Moreover, the field has expanded into micro- and nanotechnologies, with impactful contributions in electronics, mechanics, and biology.

Recent advancements in ultra-short laser pulse technology, particularly through the development of chirped pulse amplification, have enabled precise control over pulse duration—ranging from several nanoseconds down to the atto-seconds regime—tailored to specific application requirements. In light of this, the demand for more accurate and physically comprehensive theoretical models has grown significantly. The concepts and models discussed in the preceding paragraphs can be readily extended to describe radiation‒matter interactions beyond laser radiation, including those involving electron or charged particle beams. These frameworks are applicable to a wide range of solid materials, including metals, semiconductors, and dielectrics. Notably, a strong parallel exists between laser and electron beam cladding processes, as both involve localized energy deposition, rapid melting, and resolidification, governed by similar thermal and material response dynamics.

The upcoming Special Issue of Materials, entitled "The Heat Equation: The Theoretical Basis for Materials Processing", aims to gather contributions focused on modified forms of the heat equation as well as on advanced analytical and numerical methods for their solution.

The submission of full research articles, communications, and comprehensive review papers is highly welcome.

Dr. Mihai Oane
Dr. Liviu Duta
Guest Editors

Dr. Alexandra Trefilov
Guest Editor Assistant

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• two temperature model
• theoretical laser processing
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