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This chapter discusses Pigou's withdrawal to the academy at Cambridge, which did not merely indicate a changing perception of the possibilities of government action. This withdrawal also signaled that the dream to engage in efforts that directly affected welfare was fading. These developments occurred, however, during the period of Pigou's career that would prove the most productive, a period when he consolidated his reputation as the most eminent economic voice of his generation. As he stepped back from participating in issues of policy, he was ascendant as a scientist “of the chair.” It is not unlikely that Pigou himself took some satisfaction in this transformation.

Information is physical, so its manipulation through devices is subject to its own mechanics: the science and engineering of behavioral description, which is intermingled with classical, quantum and statistical mechanics principles. This chapter is a unification of these principles and physical laws with their implications for nanoscale. Ideas of state machines, Church-Turing thesis and its embodiment in various state machines, probabilities, Bayesian principles and entropy in its various forms (Shannon, Boltzmann, von Neumann, algorithmic) with an eye on the principle of maximum entropy as an information manipulation tool. Notions of conservation and non-conservation are applied to example circuit forms folding in adiabatic, isothermal, reversible and irreversible processes. This brings out implications of fluctuation and transitions, the interplay of errors and stability and the energy cost of determinism. It concludes discussing networks as tools to understand information flow and decision making and with an introduction to entanglement in quantum computing.

Electromechanics—coupling of mechanical forces with others—exhibits a continuum-to-discrete spectrum of properties. In this chapter, classical and newer analysis techniques are developed for devices ranging from inertial sensors to scanning probes to quantify limits and sensitivities. Mechanical response, energy storage, transduction and dynamic characteristics of various devices are analyzed. The Lagrangian approach is developed for multidomain analysis and to bring out nonlinearity. The approach is extended to nanoscale fluidic systems where nonlinearities, fluctuation effects and the classical-quantum boundary is quite central. This leads to the study of measurement limits using power spectrum and, correlations with slow and fast forces. After a diversion to acoustic waves and piezoelectric phenomena, nonlinearities are explored in depth: homogeneous and forced conditions of excitation, chaos, bifurcations and other consequences, Melnikov analysis and the classic phase portaiture. The chapter ends with comments on multiphysics such as of nanotube-based systems and electromechanobiological biomotor systems.

This chapter describes three molecular theories of rubber elasticity. Section 2.1 outlines the elementary theory of Kuhn and Treloar, which is of particular importance since it presents the basic elements of rubberlike elasticity in a very transparent way. Section 2.2 presents the phantom network model developed by James and Guth, and section 2.3 presents the affine network model developed by Wall and Flory. Historical aspects of the theories have been given in an article by Guth and Mark, and in a book prepared as a memorial to Guth. Finally, the major features of both theories are briefly summarized in a review. Separately, the James-Guth theory has been reviewed by Guth and by Flory, and the phantom network model of section 2.2 is based on the Flory treatment. The affine network model has been described in detail in Flory’s 1953 book. This model is described in section 2.3 by generalizing the phantom network model (as was done in one of Flory’s subsequent studies). The simple, elementary statistical theory described in section 2.1 paved the way to the current understanding of rubber elasticity. Further progress in the understanding of rubberlike systems was possible, however, only as a result of the two more precise and accurate theories: the phantom network and the affine network theories. Despite their differences, these two theories and the corresponding molecular models have served as basic reference points in this area for more than four decades. They still serve this purpose for the interpretation and explanation of experimental data. The differences between the assumptions and the predictions of the two models have led to serious disagreements during their development, as may be seen from the original papers cited earlier. The main point of disagreement was the magnitude of the front factor that appeared in the expression for the elastic free energy and the stress. For tetrafunctional networks, the James-Guth phantom network theory predicts one-half the value of the front factor obtained by the Wall-Flory affine network theory.

In the constrained-junction model presented in chapter 3, intermolecular correlations were assumed to suppress the fluctuations of junctions. According to this model, the elastic free energy of a network varies between the free energies of the phantom and the affine networks. In a second group of models, to be introduced here, there is a constraining action of entanglements along the chains that may further contribute to the elastic free energy, as if they were additional (albeit temporary) junctions. Consequently, the upper bound of the elastic free energy of such networks may exceed that of an affine network. Since the entanglements along the chain contour are explicitly taken into account in the models, they are referred to as the constrained-chain models. The idea of constrained-chain theories originates from the trapped-entanglement concept of Langley, and Graessley, stating that some fraction of the entanglements which are present in the bulk polymer before cross-linking become permanently trapped by the cross-linking and act as additional cross-links. These trapped entanglements, unlike the chemical cross-links, have some freedom, and the two chains forming the entanglement may slide relative to one other. The two chains may therefore be regarded as being attached to each other by means of a fictitious “slip-link,” as is illustrated schematically in figure 4.1. The entangled system of chains representing the real situation is shown in part (a), and the representation of two entangled chains in this system joined together by a sliplink is shown in part (b). The slip-link may move along the chains by a distance a, which is inversely proportional to the severity of the entanglements. A model based on this picture of slip-links was first proposed by Graessley, and a more rigorous treatment of the slip-link model was given by Ball et al. and subsequently simplified by Edwards and Vilgis; section 4.1 describes this latter treatment in detail. In section 4.2, we present the extension of the Flory constrained junction model to the constrained-chain model by including the effects of constraints along chains, following Erman and Monnerie. One of the newest approaches, the diffused-constraints model, is then described briefly in section 4.3.

In the first section of this chapter, the relationships between the Helmholtz free energy, the stress tensor, and the deformation tensor are given for uniaxial stress. These relations follow from the general discussion of stress and strain given in appendix C, and the notation and approach closely follow the classic treatment of Flory. The detailed forms of the stress-strain relations in simple tension (or compression) are given in the remaining sections of the chapter for the (1) phantom network, (2) affine network, (3) constrained-junction model, and (4) slip-link model. Results of theory are then compared with experiment. The effects of swelling on the stress-strain relations are also included in the discussion. It is to be noted that the stress-strain relations in this chapter are obtained by treating the swollen networks as closed systems. The conditions for such systems are fulfilled if solvent does not move in and out of the network during deformation. A network swollen with a nonvolatile solvent and subject to simple tension in air is an example of a closed system. The same network at swelling equilibrium and subjected to compression will exude some of the solvent under increased internal pressure, and is therefore not a closed system. For semiopen systems, such as those under compression, or, in general, networks stressed while immersed in solvent, a more general thermodynamic treatment is required. This situation will be taken up in the following chapter.

Segmental or molecular orientation refers to the anisotropic distribution of chain-segment orientations in space, due to the orienting effect of some external agent. In the case of uniaxially stretched rubbery networks, which will be the focus of this chapter, segmental orientation results from the distortion of the configurations of network chains when the network is macroscopically deformed. In the undistorted state, the orientations of chain segments are random, and hence the network is isotropic because the chain may undertake all possible configurations, without any bias. In the other hypothetically extreme case of infinite degree of stretching of the network, segments align exclusively along the direction of stretch. The mathematical description of segmental orientation at all levels of macroscopic deformation is the focus of this chapter. Segmental orientation in rubbery networks differs distinctly from that in crystalline or glassy polymers. Whereas the chains in glassy or crystalline solids are fully or partly frozen, those in an elastomeric network have the full freedom to go from one configuration to another, subject to the constraints imposed by the network connectivity. The orientation at the segmental level in glassy or crystalline networks is mostly induced by intermolecular coupling between closely packed neighboring molecules, while in the rubbery network intramolecular conformational distributions predominantly determine the degree of segmental orientation. The first section of this chapter describes the state of molecular deformation. In section 11.2, the simple theory of segmental orientation is outlined, followed by the more detailed treatment of Nagai and Flory. The chapter concludes with a discussion of infrared spectroscopy and the birefringence technique for measuring segmental orientation. For uniaxial deformation, the deformation tensor λ takes the form λ = diag(λ, λ-1/2, λ-1/2), where diag represents the diagonal of a square matrix, and λ is the ratio of the stretched length of the rubbery sample to its undeformed reference length. The first element along the diagonal of the matrix represents the extension ratio along the direction of stretch, which may be conveniently identified as the X axis of a laboratory-fixed frame XYZ. The other two elements refer to the deformation along two lateral directions, Y and Z.

Classical theories of rubber elasticity are based on models of flexible polymer chains that are sufficiently long to exhibit Gaussian behavior as described in chapter 1 and in appendix F. Additionally, the chains are phantomlike in the sense that they do not interact with one other along their contours. The theories described in chapter 2 were based on this picture of the individual chain. In this chapter, we describe the elasticity of networks that depart substantially from those addressed in the classical theories. The departures may result from two sources: (1) the chains may be only semiftexible, as a result of which the segments of neighboring chains compete for space in the deformed network, and choose preferentially oriented configurations, and (2) the chains may form crystallites, upon deformation, as a result of which the homogeneous structure of the classical network model may be transformed into a nonhomogeneous one having microphases of crystalline and amorphous regions. The subject of crystallization under deformation, for networks in general, is relatively old, and has been treated in some detail in previous studies. For this reason, crystallization and some of its effects will be reviewed only briefly at the end of this chapter. The main emphasis will be given to networks with semiflexible chains. Examples of networks with semiflexible chains are those in which the chains have rodlike segments separated by flexible spacers, or those where the chains have bond angles appreciably larger than tetrahedral. Incorporation of these chains into a network structure results in materials that exhibit segmental orientations significantly larger than those shown by classical networks. Specific examples would include networks prepared from aromatic polyamide chains or from chains containing liquid-crystalline sequences along the direction of the backbone. Because of their unique chain structures, these networks are easily orientable, at the molecular level, by macroscopic deformations. The orientational transitions may easily be controlled by application and removal of anisotropic strains, and are therefore of great technological interest for use in optical devices. Other examples of networks with easily orientable chains are those with rigid sequences in the side groups.

Small-angle neutron scattering (SANS) experiments from networks were initiated by Benoit and collaborators in the mid-1970s. Currently, SANS is an important major technique used in studying network structure and behavior. Its importance lies in its being a direct method with which observations may be made at the molecular-length scale without the need for a theoretical model for interpreting the data. This feature makes neutron scattering a valuable tool for testing various molecular theories on which current understanding of elastomeric networks is based. The general features of the technique are explained in section 14.1, followed in section 14.2 by a review of relevant experimental work. Section 14.3 then describes different theories of neutron scattering from networks, and compares them with experimental results. The technique of neutron scattering and its application to polymers in the dilute and bulk states, to blends, and to networks are described in several review articles and a book. The reader is referred to this literature for a more comprehensive understanding of the technique and the underlying theory. The neutrons incident on a sample during a typical experiment are from a nuclear reactor. Neutrons leaving the source are first collimated so that they arrive at the sample in the form of plane waves. Figure 14.1 shows such an incident neutron wave on two scattering centers i and j. After interacting with the scattering centers, the neutrons move in various directions. In a neutron scattering experiment, the intensity of the scattered neutron wave is measured as a function of the angle θ shown in the figure, in which the vectors k0 and k are the wave propagation vectors for incident and scattered neutron rays, respectively. In general, the magnitudes of k0 and k differ if there is energy change upon scattering, and in this case the scattering is called inelastic. Inelastic scattering experiments are particularly useful in studying the dynamics of a system, such as relaxation or diffusion.