Volume 31, Issue 4

The objective of the present work is to compare several thermomechanical frameworks, taking into account the influence of strain gradient, internal variables, gradient of internal variables, and temperature gradient on the constitutive behavior of materials. In particular, the restrictions by the second law of thermodynamics are derived. The method of exploitation consists of two steps: an application of the well-known method by Liu and a new method of exploiting the residual inequality. The first example introduces an enlarged set of variables for the constitutive functions including in particular the strain gradient, an internal variable, its gradient, and the temperature gradient. In the second example, the power of internal forces is enriched to incorporate generalized stress measures. In the third example, the classical thermomechanical setting is complemented by a balance-type differential equation for an additional variable. Finally, material theories of grade n are envisaged. It is shown that the free energy density may depend on gradients only in the case that an additional balance equation is introduced. We also demonstrate that for isotropic materials the second law of thermodynamics implies for a large class of state spaces that the entropy flux equals the heat flux divided by temperature.

Fick's first law describes diffusive flux as a linear function of the concentration gradient; its most popular generalization describes flux as a linear function of the chemical potential gradient. This generalization and others have been used for modeling, but the relationships between the flux and the gradients are nonlinear because the coefficients generally depend on state variables. In this paper, lattice density functional theory equations for diffusive flux are recast into linear differential form to explore well-known diffusion equations. This shows that the diffusive flux is related to the gradients of state variables (e.g., to the chemical potential gradient) through integrating factors, much like the heat flux in Clausius's theorem is related to the difference in entropy. Subsequent analyses show why diffusive flux is more linear with respect to the fugacity gradient than the chemical potential gradient; and that the gradient of another property has linear proportionality to the diffusive flux. This property can appear as the impingement rate onto vacancies and molecules of species whose density gradients can be influenced by diffusion; and it behaves similarly to the chemical potential and the fugacity. The results agree with the theory of non-equilibrium thermodynamics about the conjugate force vectors for the diffusive flux.

The process of die swell in polymer jets is an important feature within polymer processing and can be explained through a study of its rheological effects. The existence of a thermocapillary effect, driven by the gradient of its surface tension, should be considered when examining a thermal jet that has a non-uniform temperature distribution on its free surface, as in various polymer processings. Both the rheological effect and thermocapillary effect on die swell can be studied numerically through a finite element method as used on a two-dimensional and unsteady model, in which a Coleman–Noll second-order fluid model is employed. The results show that the expanding angle depends on both the rheological property of the fluid and the pressure at the vessel exit. Although both the thermocapillary and the rheological effects contribute to the cross-section expansion of the fluid jet, the latter is more important in determining the expansion.

Critical phenomena in non-equilibrium systems have been studied by means of a wide variety of theoretical and experimental approaches. Mode-coupling, renormalization group, complex Lie algebras and diagrammatic techniques are some of the usual theoretical tools. Experimental studies include light and inelastic neutron scattering, X-ray photon correlation spectroscopy, microwave interferometry and several other techniques. Nevertheless, no conclusive treatment has been developed from the basic principles of a thermodynamic theory of irreversible processes. We have developed a formalism in which we obtain correlation functions as field averages of the associated functions. By applying such formalism, we attempt to find out whether the resulting correlation functions will inherit the mathematical properties (integrability, generalized homogeneity, scaling laws) of its parent potentials , and we also use these correlation functions to study the behavior of macroscopic systems far from equilibrium , especially in the neighborhood of critical points or dynamic phase transitions. As a working example, we will consider the mono-critical behavior of a non-equilibrium binary fluid mixture close to its consolute point.

A model of a mixture with N different components is analyzed from a thermodynamic viewpoint with internal variables, both in the presence and absence of viscosity. Thermodiffusion phenomena are described by N internal variables ( N ≥ 2). General thermodynamic restrictions and residual dissipation inequalities are obtained by the Clausius–Duhem inequality.