Elasticity thermodynamics

Bianchi, U., and E. Pedemonte Rubber elasticity Thermodynamic properties of deformed networks. J. Polymer Sci. Pt A-2, 5039 (1964). 

Schematic derivation of the size-dependent FeP04-LiFeP04 binary phase diagram using the XRD data for Lio.6FeP04 is illustrated in Fig. 14.5, which highlights the fact that the solid solubility limits a, jS, in LiaFeP04, and Lii FeP04 systematically increases along with the reduction in particle size. Most probably, these systematic variations originate from the elastic thermodynamic effects on the...

New mathematical techniques [22] revealed the structure of the theory and were helpful in several derivations to present the theory in a simple form. The assumption of small transient (elastic) strains and transient relative rotations, employed in the theory, seems to be appropriate for most LCPs, which usually display a small macromolecular flexibility. This assumption has been used in Ref [23] to simplify the theory to symmetric type of anisotropic fluid mechanical constitutive equations for describing the molecular elasticity effects in flows of LCPs. Along with viscoelastic and nematic kinematics, the theory nontrivially combines the de Gennes general form of weakly elastic thermodynamic potential and LEP dissipative type of constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank elasticity in liquid crystalline polymers. It should be mentioned that this theory is suitable only for monodomain molecular nematics. Nevertheless, effects of Frank (orientation) elasticity could also be included in the viscoelastic nematody-namic theory to describe the multidomain effects in flows of LCPs near equilibrium. 

The equations of electrocapillarity become complicated in the case of the solid metal-electrolyte interface. The problem is that the work spent in a differential stretching of the interface is not equal to that in forming an infinitesimal amount of new surface, if the surface is under elastic strain. Couchman and co-workers [142, 143] and Mobliner and Beck [144] have, among others, discussed the thermodynamics of the situation, including some of the problems of terminology. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

Polymers with the mechanical and chemical properties we have discussed in this section are called elastomers. In the next couple of sections we shall examine the thermodynamic basis for elasticity and then apply these ideas to cross-linked polymer networks. 

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

Note this is the same derivation that yields the important results V = (3G/3p)y and S = (3G/3T)p when no elastic work is considered. It would be inappropriate for a book like this to digress into thermodynamics any further than this. The two relationships cited above are derived in almost every thermodynamics text the student is advised to consult a suitable reference and review this material if the treatment above is too abbreviated. 

Since entropy plays the determining role in the elasticity of an ideal elastomer, let us review a couple of ideas about this important thermodynamic variable ... 

The interface region in a composite is important in determining the ultimate properties of the composite. At the interface a discontinuity occurs in one or more material parameters such as elastic moduli, thermodynamic parameters such as chemical potential, and the coefficient of thermal expansion. The importance of the interface region in composites stems from two main reasons the interface occupies a large area in composites, and in general, the reinforcement and the matrix form a system that is not in thermodynamic equiUbhum. 

Foams are thermodynamically unstable. To understand how defoamers operate, the various mechanisms that enable foams to persist must first be examined. There are four main explanations for foam stabiUty (/) surface elasticity (2) viscous drainage retardation effects (J) reduced gas diffusion between bubbles and (4) other thin-film stabilization effects from the iateraction of the opposite surfaces of the films. 

To make the flaw grow, say by 1 mm, we have to tear the rubber to create 1 mm of new crack surface, and this consumes energy the tear energy of the rubber per unit area X the area of surface torn. If the work done by the gas pressure inside the balloon, plus the release of elastic energy from the membrane itself, is less than this energy the tearing simply cannot take place - it would infringe the laws of thermodynamics. 

In an appropriately designed experiment, it is possible to measure the pull-off force (Ps), contact radius (a versus P, ao and aj, and the separation profile outside the contact zone (D versus j ). From these measurements, it is possible to determine the thermodynamic work of adhesion between two surfaces, if the contacting bodies are perfectly elastic. 

Viscoelastic polymers essentially dominate the multi-billion dollar adhesives market, therefore an understanding of their adhesion behavior is very important. Adhesion of these materials involves quite a few chemical and physical phenomena. As with elastic materials, the chemical interactions and affinities in the interface provide the fundamental link for transmission of stress between the contacting bodies. This intrinsic resistance to detachment is usually augmented several folds by dissipation processes available to the viscoelastic media. The dissipation processes can have either a thermodynamic origin such as recoiling of the stretched polymeric chains upon detachment, or a dynamic and rate-sensitive nature as in chain pull-out, chain disentanglement and deformation-related rheological losses in the bulk of materials and in the vicinity of interface. 

The premise of the above analysis is the fact that it has treated the interfacial and bulk viscoelasticity equally (linearly viscoelastic experiencing similar time scales of relaxation). Falsafi et al. make an assumption that the adhesion energy G is constant in the course of loading experiments and its value corresponds to the thermodynamic work of adhesion W. By incorporating the time-dependent part of K t) into the left-hand side (LHS) of Eq. 61 and convoluting it with the evolution of the cube of the contact radius in the entire course of the contact, one can generate a set of [LHS(t), P(0J data. By applying the same procedure described for the elastic case, now the set of [LHS(t), / (Ol points can be fitted to the Eq. 61 for the best values of A"(I) and W. 

Fig. 2.10. Certain high strength solids with low thermal conductivity show a loss or reduction of shear strength when loaded above the Hugoniot elastic limit. The idealized behavior of such solids upon loading is shown here. The complex, heterogeneous nature of such yield phenomena probably results in processes that are far from thermodynamic equilibrium.

Fig. 2.15. Release wavespeeds at very high pressure can be determined by experiments in which the sample thickness is varied for fixed thickness of a high velocity impactor. Data on aluminum alloy 2024 are shown. As indicated in the figure, shear velocity (C ) and Poisson s ratio (cr) at pressure can be calculated from the elastic and bulk speeds if thermodynamic equilibrium is assumed (after McQueen et al. [84M02]).

The parameters which characterize the thermodynamic equilibrium of the gel, viz. the swelling degree, swelling pressure, as well as other characteristics of the gel like the elastic modulus, can be substantially changed due to changes in external conditions, i.e., temperature, composition of the solution, pressure and some other factors. The changes in the state of the gel which are visually observed as volume changes can be both continuous and discontinuous [96], In principle, the latter is a transition between the phases of different concentration of the network polymer one of which corresponds to the swollen gel and the other to the collapsed one. 

As a result of simultaneous introduction of elastic, viscous and plastic properties of a material, a description of the actual state functions involves the history of the local configuration expressed as a function of the time and of the path. The restrictions, which impose the second law of thermodynamics and the principle of material objectivity, have been analyzed. Among others, a viscoplastic material of the rate type and a strain-rate sensitive material have been examined. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

In the following pages I have endeavoured to deduce the principles of Thermodynamics in the simplest possible manner from the two fundamental laws, and to illustrate their applicability by means of a selection of examples. In making the latter, I have had in view more especially the requirements of students of Physical Chemistry, t6 whom the work is addressed. For this reason chemical problems receive the main consideration, and other branches are either treated briefly, or (as in the case of the technical application to steam and internal combustion engines, the theories of radiation, elasticity, etc.) are not included at all. 

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