Curved interfaces, thermodynamics

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. 

Up to this point we have dealt with the thermodynamics of planar boundaries. Let us add several relations for curved interfaces. First, we have to establish an equivalent to the Gibbs-Thomson equation which holds for curved external surfaces in a multi-component system. For incoherent (fluid-like) interfaces, this can be done by considering Figure 10-5. From the equilibrium condition at constant P and T, one has... 

The present volume involves several alterations in the presentation of thermodynamic topics covered in the previous editions. Obviously, it is not a trivial exercise to present in a novel fashion any material that covers a period of more than 160 years. However, as best as I can determine the treatment of irreversible phenomena in Sections 1.13, 1.14, and 1.20 appears not to be widely known. Following much indecision, and with encouragement by the editors, I have dropped the various exercises requiring numerical evaluation of formulae developed in the text. After much thought I have also relegated the Caratheodory formulation of the Second Law of Thermodynamics (and a derivation of the Debye-Hiickel equation) as a separate chapter to the end of the book. This permitted me to concentrate on a simpler exposition that directly links entropy to the reversible transfer of heat. It also provides a neat parallelism with the First Law that directly connects energy to work performance in an adiabatic process. A more careful discussion of the basic mechanism that forces electrochemical phenomena has been provided. I have also added material on the effects of curved interfaces and self assembly, and presented a more systematic formulation of the basics of irreversible processes. A discussion of critical phenomena is now included as a separate chapter. Lastly, the treatment of binary solutions has been expanded to deal with asymmetric properties of such systems. 

F.C. Goodrich, Thermodynamics of Fluid Interfaces, in Surface and Colloid Science, E. Matijevic, Ed., Wiley (1969), Vol. 1, p. 1. (Formal thermodjmamics, mechanics, tensors, curved interfaces.)... 

W. Actually, the surfactant does not like to be in water nor in oil because one part of the molecule is always lyophobic, which is why micelles are formed to hide it away from the solvent. Hence, it may be said that in type I phase behaviour the surfactant dislikes more oil than water, and in type II it dislikes more water than oil. Then, in type III phase behaviour, the surfactant equally dislikes both phases and would seek a third alternative, e.g. forming a bicontinuous microemulsion. In thermodynamic terms, it simply means that the chemical potential of the surfactant in such a microemulsion phase is lower than when it is adsorbed at the curved interface of a drop. 

The thermodynamic description of the interfacial state of liquid systems is the basis for the development of relationships between the adsorption density T at a liquid interface, the surface tension and the surfactant bulk concentrations. Beside this the interfacial tension is an intrinsic parameter which determines the shape of a curved interface as well as different other types of capillary phenomena. 

THERMODYNAMICS OF CURVED INTERFACES 2.3.1 Capillarity and Atom Activity... 

Consider a system where two phases, a and p, are separated by a curved interface and are in equilibrium. If the total volume V, temperature T and chemical potential (Xt are constant in this system, the change in thermodynamic potential caused by an infinitesimal movement of the interface is null. That is... 

FIGURE 73 Thermodynamic explanation for the melting point depression observed in nanoscale particles. Compared to G foo) for the hulk solid, G (r) for a finite-sized particle is slightly higher due to the excess chemical potential associated with the curved interface of the particle. Gj is not affected. This results in a measurable decrease in for small-radius particles compared to the bulk solid. 

Example 2.7 Derive the following relationships for thermodynamic functions C7, A, and G of an open system with a curved interface ... 

Thermodynamics of Curved Interfaces in Relation to the Helfrich Curvature Free Energy Approach Jan Christer Eriksson and Stig LJunggren... 

A few years later, in 1977, Boruvka and Neumann [10] published the first rigorous thermodynamic treatment of fully equilibrated, curved interfaces, resulting in a generalized Laplace equation of the kind ... 

This book chapter is complementary to a previous review article on the mechanics and thermodynamics of curved interfaces by Kralchevsky and the... 

The book begins with a review of the relevant aspects of the thermodynamics of bulk systems, followed by a description of the thermodynamic variables for surfaces and interfaces. Important surface phenomena are detailed, including wetting, crystalline systems (mcluding grain boundaries), interfaces between different phases, curved interfaces (capillarity), adsorption phenomena, and adhesion of surface layers. The later chapters also feature case studies to illustrate real-world applications. Each chapter includes a set of study problems to reinforee the reader s understanding of important concepts, with solutions available for instructors online via www.cambridge.org/meier. 

Notably, however, the energy required for emulsification exceeds the thermodynamic energy AAyi2 by several orders of magnitude [39]. This is because a significant amount of energy is needed to overcome the Laplace pressure, Ap, which results from the production of a highly curved interface (small droplets), i.e. 

Below, we first introduce the most general mechanical description of the surface moments (torques) exerted on the boundary between two fluid phases. Then, we consider the thermodynamics of a curved interface (membrane) in terms of the work of flexural deformation. Next, we specify the bending rheology by means of the model of Helfiich [202]. Finally, we review the available expressions for the contributions of the electrostatic, steric, and van der Waals interactions to the interfacial bending moment and curvature elastic moduli. These expressions relate the interfacial flexural properties to the properties of the adsorbed surfactant molecules. 

Two approaches, mechanical and thermodynamical, exist for the theoretical description of general curved interfaces and membranes. The first approach originates from the classical theory of shells and plates, reviewed in Refs. 202 and 204. The surface is regarded as a two-dimensional continuum whose deformation is described in terms of the rate-of-strain tensor and the tensor of curvature. In addition, the forces and the force moments acting in the interface are expressed by the tensors of the interfacial stresses, , and moments (torques), M. Figure 11 illustrates the physical meaning of the components of the latter two tensors. Usually, they are expressed in the form... 

For a complete mechanical description of the surface, one needs to specify expressions for the stresses and moments. This is usually done by postulating some constitutive relations between stress and strain, which pertain to a particular model for the rheological behavior of the interface. An example is the Scriven s constitutive relation [140] for a see Eq. (81). In Section IV.C, we discuss a constitutive relation for the tensor of the surface moments, M. Before that, we consider the thermodynamics of the curved interfaces. 

The basic idea of the local thermodynamic description, which is due to Gibbs [8], is to apply the fundamental equation of a uniform surface phase locally (i.e., for each elementary portion, dA, of the curved interface) ... 

In this section, we consider the problems relevant to equiUbriiim of the two multicomponent phases separated by a curved interface. This is the classical and the most well-studied case of the thermodynamic equilibrium involving surface effects. Such equilibrium is present in macro-porous and mesoporous media, like the porous rocks of petroleum reservoirs, where it accompanies adsorption. In the pores of smaller sizes, the forces produced by the solid surfaces may modify the properties of the bulk (Uquid and gas) phases. However, the present study is also important to the pores of smaller sizes, as it makes it possible to separate the effects connected with the gas-liquid surface tension (and, of course, the contact angle) from additional contributions of the solid walls. The corrections related to the last type of interactions have been considered in, for example. Refs. [13-15]. For brevity, we will apply the term capillary equilibrium to the narrow case being described, but it must be remembered, however, that a wider understanding of the capillary equilibrium is available. 

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