Configuration force

To initiate a chemical relaxation it is necessary to perturb the system from its initial equilibrium position. This is done by applying a forcing function, which is an appropriate experimental stress to which the system responds with a shift in equilibrium configuration. Forcing functions can be transient (a sudden, essentially discontinuous Jolt ) or periodic (a cyclic stress of constant frequency). 

Electrostatic. Virtually all colloids in solution acquire a surface charge and hence an electrical double layer. When particles interact in a concentrated region their double layers overlap resulting in a repulsive force which opposes further approach. Any theory of filtration of colloids needs to take into account the multi-particle nature of such interactions. This is best achieved by using a Wigner-Seitz cell approach combined with a numerical solution of the non-linear Poisson-Boltzmann equation, which allows calculation of a configurational force that implicitly includes the multi-body effects of a concentrated dispersion or filter cake. 

When the catalyst is immobilized within the pores of an inert membrane (Figure 25.13b), the catalytic and separation functions are engineered in a very compact fashion. In classical reactors, the reaction conversion is often limited by the diffusion of reactants into the pores of the catalyst or catalyst carrier pellets. If the catalyst is inside the pores of the membrane, the combination of the open pore path and transmembrane pressure provides easier access for the reactants to the catalyst. Two contactor configurations—forced-flow mode or opposing reactant mode—can be used with these catalytic membranes, which do not necessarily need to be permselective. It is estimated that a membrane catalyst could be 10 times more active than in the form of pellets, provided that the membrane thickness and porous texture, as well as the quantity and location of the catalyst in the membrane, are adapted to the kinetics of the reaction. For biphasic applications (gas/catalyst), the porous texture of the membrane must favor gas-wall (catalyst) interactions to ensure a maximum contact of the reactant with the catalyst surface. In the case of catalytic consecutive-parallel reaction systems, such as the selective oxidation of hydrocarbons, the gas-gas molecular interactions must be limited because they are nonselective and lead to a total oxidation of reactants and products. For these reasons, small-pore mesoporous or microporous... 

The notion of a configurational force is entirely in keeping with our aim of developing effective theories for characterizing the behavior of materials. In... 

The central observation associated with the definition of configurational forces is that the total energy of the body of interest and associated loading devices depends explicitly on the positions of the various defects within that body. A small excursion of a given defect from position Xj to Xj + S i will result in an attendant change of the total energy. The configurational force on that defect associated with that motion is defined via... 

As noted above, the notion of a configurational force may be advanced as a basis for considering the dynamics of defects themselves since, once such forces are in hand, the temporal evolution of these defects can be built up in turn by the application of an appropriate kinetic law which postulates a relation of the form V = v(driving force). 

To further elaborate the underlying idea of a configurational force, we appeal to the examples indicated schematically in fig. 2.8. Fig. 2.8(a) shows an interface within a solid and illustrates that by virtue of interfacial motion the area of the interface can be reduced. If we adopt a model of the interfacial energy in which it is assumed that this energy is isotropic (i.e. y does not depend upon the local interface normal n), the driving force is related simply to the local curvature of that interface. Within the theory of dislocations, we will encounter the notion of image dislocations as a way of guaranteeing that the elastic fields for dislocations in finite... 

Fig. 2.8. Representative examples of the origins of the concept of configurational forces (a) curved interface, (b) dislocation near a free surface and (c) solid with a crack.

Mathematical Analysis in the Mechanics of Fracture by James R. Rice, from Vol. II of Fracture edited by G. Sih, Academic Press, New York New York, 1968. Rice s account of fracture remains, in my opinion, one of the definitive statements of many of the key ideas in the subject. Section E on Energy Variations and Associated Methods is especially pertinent to our discussion on configurational forces and gives an in depth application of these ideas in the fracture context. 

This equation is written on the basis of the observation that the infinitesimal element of area swept out by the dislocation is characterized by the vector product dy X d. Note that dy is the local excursion of the segment, while d is a vector along the line at the point of interest. If we now recall that the configurational force is given as SEm = —Em m, then we may evidently write the configurational force in the present context as... 

Note that in contrast to the result in eqn (8.50), there is a factor of 4 rather than 2 in the denominator. This arises because the separation of the image dislocations themselves is 2d, not d. We have now seen that by virtue of the presence of a free surface, an image force is induced on a subsurface dislocation. In addition, the idea of an image force has further reinforced our use of configurational forces and their origins in the breaking of translational symmetry. 

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