Reference system, change symmetrical

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

Numerous methods for measuring fluid viscosity exist, for example, capillary tube flow methods (Ostwald viscometer), Zahn cup method, falling sphere methods, vibrational methods, and rotational methods. Rotational viscometers measure the torque required to turn an object immersed or in contact with a fluid this torque is related to the fluid s viscosity. A well-known example of this type of system is the Couette viscometer. However, it should be noted that as some CMP slurries may be non-Newtonian fluids, the viscosity may be a function of the rotation rate (shear rate). An example of this is the dilatant behavior (increasing viscosity unda increasing shear) of precipitated slurries that have symmetrical particles [33]. Furthermore, the CMP polisher can be thought of as a large rotational plate viscometer where shear rates can exceed 10 s and possibly affect changes to the apparoit viscosity. The reader can refer to the comprehensive review of viscosity measurement techniques in the book by Viswanath et aL [34]. 

Fife refers to a set of partial differential equations like eqs. (6.28) as a propagator—controller system, for reasons that will become clear later. We recall that the small parameter e 1 makes u change much more rapidly than v. It also makes the time scale of the chemical change much shorter than that of the diffusion that is, the diffusion terms are small compared with the chemical rates of change. Note that all the chemistry is in the functions / and g, which are constructed so that neither concentration can grow to infinity. We will want to consider either a one-dimensional geometry (spatial variable x) or a radially symmetric two-dimensional system, like a Petri dish. 

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