This open access textbook fills a gap, in that it introduces readers to the theory and applications of the Phase-Field technique. Phase Field, over the years, has emerged as a standard tool for materials research, just as the Finite-Element technique has in structure mechanics.  Whereas the few existing textbooks on this topic are intended for advanced readers, this one is made accessible to the widest possible audience, through an engaging, lecture format. The content grows out of a course the authors teach for graduate students at Ruhr-University Bochum. Even readers who may, at first, have no clue at all what a “Phase Field” is and for what it is used, are invited on a journey from general physics of thermodynamics and wave mechanics, through applications in all fields of materials science, up to the central questions of physical being. On this journey all the necessary techniques are detailed, mostly formulated in a mathematical language easily understood by engineers and natural scientists.

Prof. Dr. Ingo Steinbach is a pioneer of the phase-field theory with early publications from 1996. His Multi-Phase-Field approach, with contributions from many other scientists worldwide, forms a standard for multi-phase application in metallurgy and other fields of materials research. Applications range from pattern formation during solidification over solid-state transformation during materials processing to materials degradation and failure. He recently went back to the roots of the phase-field theory in non-linear wave mechanics to propose a Quantum Phase Field Concept of Matter, which forms a new paradigm for understanding the physical world.

Hesham Salama is a Research Associate and a PhD student under the supervision of Prof. Steinbach in the context of the phase-field theory. He is one of the developers of the OpenPhase software project targeted at numerically maximum efficient phase-field simulations.