Thermodynamics of microstructure evolution: grain growth

It is gradually getting clear that the macroscopic description of microstructure evolution requires additional thermodynamic parameters, entropy of microstructure and temperature of microstructure. It was claimed that there is "one more law of thermodynamics": entropy of microstructure must decay in isolated thermodynamic stable systems. Such behavior is opposite to that of thermodynamic entropy. This paper aims to illustrate the concept of microstructure entropy by one example, the grain growth in polycrystals. The grain growth is treated within the framework of a theory which is a modi…cation of Hillert theory. The modi…cation is made in order to reach simultaneously two goals: to get a coincidence of theoretical predictions with experimentally observed results and to obtain the equations that admit analytical solutions. Due to these features, the modi…ed theory is of independent interest. In the modi…ed Hillert theory one observes the decay of total microstructure entropy when the system approaches the self-similar regime. The microstructure entropy per one grain grows indicating a chaotization of grain sizes. It is shown also that there exits an equation of state of grain boundary microstructure that links entropy of microstructure, energy of microstructure, average grain size and a characteristic of the inhomogeneity of the large grain distribution. Keywords: microstructure entropy, con…gurational entropy, e¤ective temperature, grain growth I. THERMODYNAMICS OF MICROSTRUCTURE EVOLUTION By thermodynamics of microstructure one means a macroscopic description of bodies possessing an evolving microstructure at a mesoscopic level. To describe a microstructure evolution within the framework of classical thermodynamics, one has to specify a set of macroscopic parameters characterizing the material and the microstructure, 1; :::; k; and the dependence of energy E on these parameters and on thermodynamic entropy S: E = E (S; 1; :::; k) : (1) Then the governing evolution equations that respect thermodynamic laws follow from the usual thermodynamic formalism. Applying this procedure to bodies with evolving microstructures, one faces the following problem: energy of the microstructure is not a function of 2 macroscopic parameters of the microstructure and must be treated as an independent additional characteristic of the microstructure1. Consider why this occurs for the process of grain growth in polycrystals. If temperature of a polycrystal is increased to 0:2 0:3 of melting temperature, the grains begin to grow. The simplest macroscopic parameter that describes the state of the grain boundary structure is the average grain size, R. It is introduced usually in the following way: one measures the number of grains in the polycrystal, N , and its volume, jV j, and de…ne R as 2R = jV j N 1=3 : (2) Formula (2) corresponds to envisioning the average grain as a cube with the side 2R; if it is a sphere of radius R (the usual choice), a numerical factor should be included in (2). Suppose that grain boundaries are the only defects of crystal structure. Then energy of the polycrystal is the sum of energy of heat motion of atoms, E0; and energy of grain boundaries, Em; (subscript m stands for microstructure): E = E0 (S) + Em: (3) If Em were a certain function of R, then the usual thermodynamic formalism follows. This, however, is not the case. Indeed, let all grain boundaries are high angle boundaries, and, thus, the total energy of grain boundaries is the product of a constant, the grain boundary energy per unit area, ; and the total area of grain boundaries. Then from dimension reasoning energy of microstructure per unit volume, Um = Em /jV j ; can be written as Um = Em jV j = X R ; (4) where X is a dimensionless parameter. It has the meaning of dimensionless boundary area, X = grain boundary area average grain size total volume (5) or dimensionless grain boundary energy, X = grain boundary energy average grain size total volume : 1 This point has been made in (Berdichevsky 2005, 2008). 3 FIG. 1: A grain boundary structure with dimensionless boundary area (6). If X were a universal constant, then the evolution of grain boundaries can be treated within the usual thermodynamic formalism. However, X does depend on microstructure. It is seen if we …nd X for various microgeometries. For example, for tessellation of space in cubes of equal sizes, X = 1:5, for Poisson-Voronoi tessellation (Ohser et al. 2000) X is about the same, X = 1:45, but not precisely the same. Larger deviations one gets, for example, for geometry shown in Fig. 1. The dimensionless boundary area X decreases as the ratio a2=a1 increases: as easy to see, X = 1:5 1 + 2 1 + 3 2 1=3 ; = a2 a1 : (6) In the limit a2=a1 !1 X tends to 0.94. These examples show that X and R and, thus, Um and R are independent. The independence of X and R can be seen also from consideration of one grain: volume of any grain and area of its boundary are independent. One can …x the grain boundary area and change the grain volume. The grain volume and grain boundary area are linked only by the isoperimetric inequality. The same conclusion can be drawn from studying the statistics of grain microstructures (Glicksman 2005, Glicksman 2005a, Graner et al. 2000). So the evolving grain microstructure possesses at least two independent macroscopic characteristics, Um and R. Let us consider the simplest case when Um and R are the only characteristics needed in a macroscopic theory, i.e. grain boundary structure is a two-parametric system (we ignore for a moment thermodynamic entropy or temperature assuming that the grain growth occurs either at adiabatic or at isothermal conditions). This case does not …t the usual thermodynamic 4 scheme because Um is not a certain function of R. However, we can set an analogy with usual thermodynamic systems. The simplest example of thermodynamic system with two independent parameters is an ideal gas. The ideal gas is characterized by temperature T and mass density : In this case the physical properties of the gas are described by the dependence of free energy on T and : One can choose energy per unit mass, U , and as the independent thermodynamic parameters of the gas. For such a choice, the physical properties of the gas are described by the equation of state