Thermodynamics of Stressed Solids

A defining characteristic of a solid is the ability to resist shear. Therefore, stress is an additional feature which has to be taken into account when the physical chemistry of solids is at issue. Gibbs treated the thermodynamics of stressed solids a century ago in his classic work Equilibrium of Heterogeneous Substances under the title The Conditions of Internal and External Equilibrium for Solids in Contact with Fluids with Regard to all Possible States of Strain of the Solid . We have already mentioned in the introduction that stress is an unavoidable result of chemical processes in solids. Let us therefore briefly discuss the basic concepts of the thermodynamics of stressed solids. 

To this end, we consider the thermodynamic functions of a homogeneously stressed solid, e.g., [L.D. Landau, E.M. Lifshitz (1989) W. W. Mullins, R. Sekerka (1985)]. In contrast to the unstressed solid, the internal energy of which is U(S, K ,), the internal energy of a stressed solid is given as U(S, VuJk,nj). For the total differential of the internal energy one has1 

1 The convention for the summation of repeated vector and tensor indices is followed here. The summation goes over 1,2,3. 

From Eqn. (14.3), one can easily derive the cross relations between extensive and intensive state variables as, for example, 

By expanding the Helmholtz free energy F at constant T in an arithmetic series in terms of ujk, we see that the linear terms vanish in view of the equilibrium condition ( rJk = 0 for unstressed crystals). Thus, from the Euler relation for homogeneous functions of second order, F is given as 

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Thermodynamics of Stressed Solids with Only Immobile Components... 

Larche, F.C. (1988) Thermodynamics of Stressed Solids, Solid State Phenomena, Vol. 3-4, 205... 

Let us investigate the steady state behavior of multicomponent crystals exposed to uniform but non-hydrostatic stresses. We first introduce some ideas on the thermodynamics of such solids (which will be discussed in more detail in Chapter 14). Solid state galvanic cells can be used to perform the appropriate experiments. 

The second type of work to which wc shall restrict our di.scussion is mechanical work. As wc shall see later, confined phases can be exposed to two types of mechanical work, namely compression (dilation) and shear. In that regard, confined phases have a lot in common with bulk solids in that they are generally inhomogeneous and anisotropic in one or more spatial dimensions. Therefore, it seems sensible to cast the mechanical work term in terms of stress (t) and dimensionless strain tensors (bulk solids [12], by writing... 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

Finally, we turn from solutions to the bulk state of amorphous polymers, specifically the thermoelastic properties of the rubbery state. The contrasting behavior of rubber, as compared with other solids, such as the temperature decrease upon adiabatic extension, the contraction upon heating under load, and the positive temperature coefficient of stress under constant elongation, had been observed in the nineteenth century by Gough and Joule. The latter was able to interpret these experiments in terms of the second law of thermodynamics, which revealed the connection between the different phenomena observed. One could conclude the primary effect to be a reduction of entropy... 

Chemists and physicists must always formulate correctly the constraints which crystal structure and symmetry impose on their thermodynamic derivations. Gibbs encountered this problem when he constructed the component chemical potentials of non-hydrostatically stressed crystals. He distinguished between mobile and immobile components of a solid. The conceptual difficulties became critical when, following the classical paper of Wagner and Schottky on ordered mixed phases as discussed in chapter 1, chemical potentials of statistically relevant SE s of the crystal lattice were introduced. As with the definition of chemical potentials of ions in electrolytes, it turned out that not all the mathematical operations (9G/9n.) could be performed for SE s of kind i without violating the structural conditions of the crystal lattice. The origin of this difficulty lies in the fact that lattice sites are not the analogue of chemical species (components). 

J.W. Cahn s early contributions to elastic coherency theory were motivated by his work on spinodal decomposition. His subsequent work with F. Larche created a rigorous thermodynamic foundation for coherency theory and stressed solids in general. A single volume, The Selected Works of John W. Cohn [15], contains papers that provide background and advanced reading for many topics in this textbook. This derivation follows from one in a publication included in that collection [16]. 

Steinicke and Linke [17] refer to several microscopic and macroscopic states of mechanically stressed solids. Short time effects can be described by stochastic means or nonequilibrium thermodynamics. Long-lasting effects can be measured by calorimetry. The chemical potential and activity of the stressed solid can be measured depending on the induced defects. These defects include ... 

Surface stress measurement — The understanding of the thermodynamics of solid/liquid interfaces is of im-... 

Such effects make quantitative studies of these re Kctions very difficult but by no means less interesting. Another difficulty associated with all studies of the reactions of solids is the dependence of the reaction rate on the previous history of the solid. A major contribution to such erratic behavior is the property of solids of being able to exist for long periods of time in metastable states of physical stress. This introduces into the description of solids an additional set of thermodynamic variables which are not necessarily at the disposal of the experimenter orSey observable. 

In our concluding remarks we can emphasize that depending on the nature of interactions between the components that constitute the medium and the solid, as well as on a combination of external conditions, one may observe the effects of various types and intensity. These include the facilitation of plastic flow of solids, or, alternatively, brittle fracture due to the action of lowered stresses mechanochemical phenomena in the zone of contact mechanically activated corrosion (the stress corrosion) the processes that are close to the spontaneous dispersion (the so-called quasi-spontaneous dispersion), and the true spontaneous dispersion, leading to the formation of thermodynamically stable lyophilic system. A great variety of types of interactions that exist between the stressed solids and the medium in contact with it requires careful and thorough examination of conditions under which... 

Contrary to this general notion that dispersions of finely divided solid particles cannot be thermodynamically stabilized, Stol and de Bruyn [34] discussed the conditions necessary for obtaining a thermodynamically stable dispersion (TSD) of solid particles in an aqueous solution medium. They stressed the role of the adsortion of potential-determining ions in lowering the interfacial free energy y to promote spontaneous dispersion and subsequently reasoned that for simple inorganic solids a decrease in y by about 200 mN/m relative to its value at the PZC may be sufficient to yield a TSD. However, the existence of thermodynamically stabilized dispersions of inorganic solids has not yet been demonstrated experimentally. 

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