As the temperature is decreased, the attractive interaction between atoms within each alloy component causes separation of the components (right image). At higher temperatures, the alloy components are randomly mixed (left image) due to thermal population of quasi-degenerate configurational states. (C) Illustration of an order disorder phase transition for a binary alloy. In thermodynamic equilibrium, the concentration of defects minimizes the free energy. The white curve shows the dependence of the Gibbs free energy of the system with the defect concentration. A generic atomic/molecular lattice model (e.g., of a crystal surface) with missing atoms/molecules illustrates a random distribution of vacancy defects. A (very) small amount of oxygen can be present in the bulk even at low pressures, while some O atoms may be missing once the oxide is formed at higher pressures. (A) Illustration of the oxidation of a metal surface as the oxygen pressure is increased, where first a change in the oxygen content at the surface and then subsequently in the bulk metal is observed. Pictorial demonstration of phenomena driven by thermodynamics. An additional important component in determining the concentration of point defects and order-disorder phase transitions is the configurational entropy, which is associated with the degeneracy of different atomic/molecular configurational states (see Figure 1).įigure 1. For example, vibrational and electronic states give rise to the corresponding entropic contributions in solid-state materials, whereas rotational and translational contributions are important for a gas. The distinguishable states of a material contributing to entropy can vary in origin, as they correspond to different degrees of freedom. The entropy can be viewed as a measure of the (quasi)degeneracy of the states of a system that are accessible at a given temperature. This effect can be formalized by introducing the concepts of entropy and free energy: at a finite temperature, the system tends to minimize its free energy rather than its internal energy because entropy is maximized. In this case, for purely statistical reasons, the system will tend to spend most of its time (i.e., will have a high probability to be found) in the parts of the PES with many local minima of a similar energy (i.e., with a high density of states), provided the energies of these states are not much higher than the global minimum. A more realistic scenario, however, is a case where the material interacts with a heat bath (a practically infinite energy reservoir, e.g., Earth's atmosphere), which keeps the temperature of the system constant. In the hypothetical situation when the system is efficiently cooled down to T = 0 K, it will eventually relax from an arbitrary state to a local or global minimum on the PES, minimizing the internal energy at these conditions. If the barriers are not very high and/or the time for the system to explore its configurational space is sufficiently large, the system will end up in a state of thermodynamic equilibrium. The system (“material plus environment”) therefore constantly samples its configurational space with a finite probability to eventually overcome barriers that separate the minima on the potential-energy surface (PES). Moreover, a material is almost always in contact with a gas (or liquid), and can exchange particles with its environment. The cluster expansion method is therefore also discussed as a numerically efficient approach for evaluating these energies.Īt finite temperatures ( T > 0 K), where functional materials typically operate, atoms move randomly in all directions due to the energy provided by heat sources. In particular, we describe approaches for calculating the configurational density of states, which requires the evaluation of the energies of a large number of configurations. We also introduce and discuss methods for calculating phase diagrams of bulk materials and surfaces as well as point defect concentrations. We demonstrate how these concepts can be used to predict the behavior of materials at realistic temperatures and pressures within the framework of atomistic thermodynamics. In this contribution, we discuss the main concepts behind equilibrium statistical mechanics. In order to understand the properties of materials at realistic conditions, statistical effects associated with configurational sampling and particle exchange at finite temperatures must consequently be taken into account. In most applications, functional materials operate at finite temperatures and are in contact with a reservoir of atoms or molecules (gas, liquid, or solid). 2Skolkovo Innovation Center, Skolkovo Institute of Science and Technology, Moscow, Russia.1Department of Chemistry and Biochemistry, University of South Carolina, Columbia, SC, United States.
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