The thermodynamics of deformation

The change in internal energy during deformation dC/ is given by 

When an elastic solid of initial length / is extended uniaxially under a tensile force /, the work done on the solid in an infinitesimal displacement is 

It is convenient to introduce the Helmholtz free energy A, which relates to changes that occur at constant volume 

for a change that occurs at constant temperature and constant volume we have 

By combining Equations (3.3) and (3.7) we see that dA = dW for isothermal changes at constant volume. Hence the tension / is given by 

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A very important problem in the thermodynamics of deformation of condensed systems is the relationship between heat and work. From Eqs. (2) and (4) by integration, the internal energy and enthalpy can be derived. As in other condensed systems, the enthalpy differs from the internal energy at atmospheric pressure only negligibly, since the internal pressure in condensed systems P > P. Therefore, the work against the atmospheric pressure can be neglected in comparison with the term jX.. Hence it follows that... 

The thermodynamics of deformation of solid-oriented polymers in any direction is determined by expressions which may be obtained from Eq. (22)7). The coefficient p may be determined by Eq. (111) and the modulus of elasticity E can be calculated according to the following expression 193)... 

The previous section described the thermodynamics of deformation and fracture in terms of the energy required to elongate and break the sample. This section describes two major experiments to evaluate the deformation and fracture energy stress-strain and impact resistance. In a tensile stress-strain experiment, the sample is elongated until it breaks. The stress is recorded as a function of extension. Stress-strain studies are usually relatively slow, of the order of mm/s. Impact strength measures the material s resistance to a sharp... 

The full significance of these observations could not be appreciated in advance of the formulation of the second law of thermodynamics by Lord Kelvin and Clausius in the early 1850 s. In a paper published in 1857 that was probably the first to treat the thermodynamics of elastic deformation, Kelvin showed that the quantity of heat Q absorbed during the (reversible) elastic deformation of any body is related in the following manner to the change with temperature in the work — TFei required to produce the deformation ... 

Lipid membranes are quite deformable, allowing water and head groups into their interiors when perturbed. A "water defect" is shown in Figure 1C, where water and lipid head groups enter the hydrophobic interior of only one of the bilayer leaflets. Figure ID shows a "water pore," where both leaflets are perturbed. At the molecular level, pore and defect formation are directly related to specific lipid-lipid interactions. It is important to understand the free energy required for pore formation in membranes and the effect of lipid composition on the process. In Section 3 of this chapter, we review recent MD studies of the thermodynamics of pore formation. 

The thickness of the interphase is a similarly intriguing and contradictory question. It depends on the type and strength of the interaction and values from 10 Ato several microns have been reported in the hterature for the most diverse systems [47,49,52,58-60]. Since interphase thickness is calculated or deduced indirectly from some measured quantities, it depends also on the method of determination. Table 3 presents some data for different particulate filled systems. The data indicate that interphase thicknesses determined from some mechanical properties are usually larger than those deduced from theoretical calculations or from extraction of filled polymers [49,52,59-63]. The data supply further proof for the adsorption of polymer molecules onto the filler surface and for the decreased mobility of the chains. Thermodynamic considerations and extraction experiments yield data which are not influenced by the extent of deformation. In mechanical measurements, however, deformation of the material takes place in all cases. The specimen is deformed even during the determination of modulus. With increasing deformations the role and effect of the immobilized chain ends increase and the determined interphase thickness also increases (see Table 3) [61]. 

Substitution of Equations (36) and (37) into Equation (35) generates a complicated differential equation with a solution that relates the shape of an axially symmetrical interface to y. In principle, then, Equation (35) permits us to understand the shapes assumed by mobile interfaces and suggests that y might be measurable through a study of these shapes. We do not pursue this any further at this point, but return to the question of the shape of deformable surfaces in Section 6.8b. In the next section we examine another consequence of the fact that curved surfaces experience an extra pressure because of the tension in the surface. We know from experience that many thermodynamic phenomena are pressure sensitive. Next we examine the effecl of the increment in pressure small particles experience due to surface curvature on their thermodynamic properties. 

The thermomechanical data accumulated during the last years as a consequence of the improvement of the technique of deformation calorimetry contributed significantly to the understanding of thermodynamics and mechanisms of the reversible deformation of polymers in the glassy, semicrystalline and rubbery state. 

In this section we consider the simplest approach to the thermodynamics of a deformed network, for which a tensor of displacement gradients is given by... 

The development of the thermodynamics of thin films is related to the problem of stability of disperse systems. An important contribution to its solving are the works of the Russian scientists Derjaguin and Landau [1] and the Dutch scientists Verwey and Overbeek [2], known today as the DVLO theory. According to their concept the particular state of the thin liquid films is due to the change in the potential energy of molecular interaction in the film and the deformation of the diffuse electric layers. The thermodynamic characteristic of a state of the liquid in the thin film, as shown in Section 3.1, appears to be the dependence of disjoining pressure on film thickness, the n(/t) isotherm. The thermodynamic properties of... 

The importance of FIPI is twofold. It can be used to promote phase inversion without changing the thermodynamics of the system to obtain a higher entropy state, or it is possible to delay phase inversion while reducing the system entropy. The characteristics of the microstructure formed (such as emulsion droplet size) are dependent on the type of microstructure and deformation (shear, extension, or combined), as well as the deformation rate. To maximize the fluid micro-structure/flow field interactions, the flow fleld must be uniform, which requires the application of the flow field over a small processing volume, which can be achieved by using MFCS mixers or CDDMs. 

Theorem A.5.5 (which is algebraic only) may be applied to the thermodynamics of our book, namely in the admissibility principle used on the models of differential type as we show in the examples below. The X are here the time or space derivatives of deformation and temperature fields other than those contained in the independent variables of the constitutive equations and therefore al a, /3, Aj, Aj, Bj are functions of these independent variables. Constraint conditions (A.99) usually come from balances (of mass, momentum, energy) and (A. 100) from the entropy inequality. 

In addition to the mechanical forms of work discussed above, there are many other forms. For example, work is involved in electrical charging that results from a current flow, in changes of surface area that are opposed by surface tension, and in magnetization caused by a magnetic field. Such forms are all equivalent to mechanical work. However, in the thermodynamics of fluids, the most common form is the mechanical work that deforms the system boundary and thereby changes the system volume. 

Although the glass transition resembles characteristics of a second-order thermodynamic transition such as changes in the coefficient of expansion and heat capacity, the temperature of the transition is a function of the heating or cooling rate and of the rate of deformation. The methods used to determine Tg are based either on static or dynamic mechanical processes. The former uses volume effects (dilatometry) and heat capacity effects in differential scanning calorimetry (DSC), entailing conditions of very low deformation. The latter utilizes the response to imposed deformation of the system. 

Under deformation, sections of chains between the network points are stretched. In a good approximation, elastomers do not change their volume under strain (Av = 0). Hence, the work of deformation (stretching) of a specimen is due only to the action of the applied force,/ According to 1st and 2nd Law of Thermodynamics, it might be separated in an energetic and entropic contribution ... 

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