Opinion: Chemical potential and materials genome

The Materials Genome Initiative started almost exactly a dozen years ago in the US.¹ However, if one asks the question “what is materials genome?” to 100 people, it is a good bet that one would get 100 different answers. To the author's knowledge, there is still no generally agreeable definition of what materials genome is, unlike from the human genome, which has a much clearer definition.² In some way, the Materials Genome Initiative has been mainly used as a rallying slogan for advocating multiscale modeling, closed-loop integration, and iteration between computation and experiments, and more recently, the applications of data science, machine learning, and artificial intelligence to materials science and engineering.^{3, 4} Therefore, the main purpose of this article is to offer the author's perspective on what could be considered as the materials genome.

Gibbs defined a simple system as one without interfacial, gravitational, electrical, and magnetic contributions and introduced a set of basic thermodynamic variables to describe an equilibrium state of a simple system.

An equilibrium state is defined as a state in which all the state variables no longer vary with time. However, it should be noted that an equilibrium state does not have to be a stable state; it can be stable, metastable, or unstable. An unstable equilibrium state is intrinsically unstable with respect to any small fluctuations in the state variables, whereas a metastable equilibrium state is stable against small fluctuations in state variables but unstable with respect to large fluctuations. A stable equilibrium state is stable against any fluctuations, large or small. Therefore, unstable and metastable equilibrium states can only be arrested kinetically in practice. However, we hypothesize that all states, including unstable and metastable states, can be described by the same set of basic state variables and the fundamental equation of thermodynamics.

These independent variables are called the natural variables for U because only when U is expressed as a function of its n + 2 natural variables, Equation (1) is a fundamental equation of thermodynamics.

Therefore, we have 2n + 5 total basic thermodynamic variables, which are related by n + 3 equations with n + 2 of them being equations of state (Equation 4) and one integrated fundamental equation of thermodynamics (Equation 1). Therefore, the number of independent variables, given by 2n + 5 − (n + 3) = n + 2, is often called the number of degrees of freedom for a system.

Here, it should be noted that x₁ + x₂ + … + x_n = 1; there are only n − 1 independent composition variables. All the thermodynamic properties of a material can be determined from the knowledge of chemical potential μ as a function of temperature, pressure, and chemical composition, x₁, x₂, …, and x_n. All the macroscopic thermodynamic properties derived from the fundamental equation of thermodynamics in terms of chemical potential are size-independent.

Similar to the chemical potential of vacancies, sum of chemical potential of electrons and chemical potential of electron holes is zero at thermodynamic equilibrium. However, it is nonzero if the electron and hole concentrations are not their equilibrium values at a given thermodynamic condition, for example, the photo-excited electrons and holes in photovoltaic devices.

Of course, can be highly nonlinear with respect to .

For an inhomogeneous system with interfaces, the inhomogeneous fields σ_ij, E_i, H_i also contribute to its thermodynamics and stability. These fields can, in principle, be solved under a set of mechanical, electric, and magnetic boundary conditions, and the corresponding strain energy, electrostatic energy, magnetic energy of an inhomogeneous system can be obtained. In principle, all the microstructures and thus properties of a material can be modeled and predicted using computational methods, such as the phase-field methods. However, as mentioned above, the quantitative prediction of all the history-dependent material properties of a practical material requires the availability of kinetic mobility data on both transport and interfacial migration as well as the energetics of interfaces that are much more challenging to experimentally measure or theoretically compute. Furthermore, the properties of a material may not only be history-dependent but also dependent on size and geometry of a material and can be highly nonlinear.

This is an opinion piece to argue that the fundamental equation of thermodynamics of a material can be considered as a history-independent Materials Genome. Therefore, one of the main goals for MGI could be to develop and build the fundamental equation for a material system. One can obtain all the thermodynamic properties from such fundamental equation, or the genome of a material, and hence there is no need to have separate databases for mechanical, thermal, and dielectric properties. With the inclusion of the order parameters describing the extents of possible internal processes, a fundamental equation also contains the thermodynamic driving forces for nonequilibrium processes and can be employed to construct stable equilibrium phase diagrams. Furthermore, all the thermodynamic information about defect energetics and electronic energy density distributions can be coded into the fundamental equation of thermodynamics. In principle, all the information that is necessary to construct a fundamental equation of thermodynamics can be computationally obtained alone. The fundamental equation of thermodynamics can also be employed in theoretical models and computational studies of phase transformations and microstructure evolution. It should be emphasized that the fundamental equation of thermodynamics can only provide partial information, that is, the thermodynamic contribution, to kinetic properties, such as chemical diffusivity, and therefore, it is not sufficient to describe history-dependent properties. Purely kinetic properties and history-dependent properties are much more challenging to obtain computationally or experimentally, and hence, it may likely have to rely on a combination of experiments, computation, and machine learning of experimental and computational data.

Long-Qing Chen: Conceptualization; writing - original draft.

There are no conflicts of interest.