Thermodynamics of Grain Boundary Segregation and Applications to Anisotropy, Compensation Effect and Prediction

Abstract

Interfacial segregation has long been studied because of its impact on various properties of materials. Usually, theoretical descriptions are based on thermodynamic terms such as segregation enthalpy and entropy. Comparisons of published papers of different authors reveal fundamental differences in our understanding and application of these thermodynamic characteristics. To clarify this situation, the report reviews several types of segregation isotherms and their derivations. Individual sets of thermodynamic state functions appearing in segregation isotherms—Gibbs energy, enthalpy and entropy—are interpreted in detail. We identify the sources of controversial interpretations and the physical meaning of particular sets of thermodynamic state functions. It is shown that for a correct interpretation an unambiguous and systematic use of the term excess in thermodynamic terminology is necessary, as well as a correct evaluation scheme for data obtained with Auger Electron Sepctroscopy (AES). Successful applications of segregation isotherms based on standard thermodynamic state functions of segregation are demonstrated for three examples: anisotropy of grain boundary segregation and its consequences for the classification of grain boundaries, compensation effect applied to grain boundary segregation, and quantitative prediction of grain boundary segregation (including anisotropy).

Keywords:

• anisotropy
• compensation effect
• grain boundaries
• prediction
• excess Gibbs energy

ACKNOWLEDGMENT

The authors gratefully acknowledge the support of the Czech Science Foundation (grants Nos. 106/05/0134 and 202/06/0004) and of the Institute Research Plan AV0Z 10100520.

Notes

* 45°[100] is an incommensurable misorientation relationship: as the ratio of the Miller indices k _s /k _l = √2, no rational k _s and k _l indices can be found for the symmetrical tilt grain boundary.

*(defined by Eq. 63) are representative of individual types of grain boundaries and were taken from Ref. ⁷⁰

a. Because there seems to be no standardized expression to subsume the terms Gibbs energy (formerly also called Gibbs free energy or free enthalpy ⁶ ), internal energy, enthalpy, entropy, volume etc., different expressions are used by different authors. For example, the Penguin dictionary of physics uses thermodynamic functions, properties or potentials. ⁷ The less specific but simple term quantities is used by Gibbs, ³ J. Cahn, ⁴ Sutton and Balluffi, ³ Zangwill, ⁸ and Wynblatt and Chatain, ⁹ and it also appears in the recommended IUPAC terminology. ⁶ Besides, J. Cahn ⁴ uses thermodynamic properties and Everett, ¹⁰ Guggenheim, ¹¹ Sparnaay, ¹² J. Cahn ⁴ and Gaskell ¹³ use functions. Flewitt and Wild ¹⁴ call them thermodynamic parameters while Sutton and Balluffi ¹ and Gaskell ¹³ use the expression state variables. In order to emphasize a thermodynamically rigorous approach (and refraining from the term thermodynamic potentials that is too easily mixed up with chemical potentials), we adopt the term thermodynamic state functions in the usage of the classic book by Lewis and Randall. ¹⁵ Because of this somewhat lengthy expression, we abridge the term to thermodynamic functions or sometimes simply refer to functions where no misunderstanding is possible.

b. Note that this “excess” term is general and not limited to interfaces.

c. Note that in the definition of the interfacial excess properties (i.e., thermodynamic functions) the terms real and ideal are used inconsistently with thermodynamics and are not defined at all.

d. Note that Δ S _I ⁰ includes all entropy contributions of the standard states (i.e., resulting from μ _i ⁰), i.e., excluding the configuration term.

e. Note that with increasing interfacial solute enrichment above a few atomic percent the ratio of the measured peak intensity of the solute to that of the matrix deviates from being proportional to solute enrichment.

f. Frequently, the compensation effect is represented as the linear dependence between the characteristic enthalpy and the pre-exponential factor of an Arrhenius relationship.

g. It is generally accepted that highly segregated general grain boundaries are broken during the fracture, which is necessary to open the interfaces for AES analysis.