Molar friction species

Classical thermod5mamics and theories of state equilibrium show an admirable flexibility with regard to the choice of components primarily only the number of these is essential. This advantage has been taken over by the diffusion theory with the thermodynamic factor as an intermediate link. As a consequence, it must be admitted that the molar frictions contained in the theory ( /c<) do not necessarily correspond to the frictional coefficients of the special molecular species contained in the mixture. So that the latter shall be the case, a component must consist of only one kind of molecule, and (in calculating the molar properties contained in the theory) the molecular weight of the component must be chosen according to the actual molecular species. 

Articles by the present writer > > and by Ljunggren are concerned with the thermod5mamics and frictional properties of components consisting of a nximber of molecular species in rapid equilibrium with one another. One result is a simple law which is vahd for the molar friction of a binary system, e.g. O of component 2, expressed in terms of the frictions of the different molecular species of this component. Assume that the molecular weight of the component is chosen to be Af, corresponding to simple molecules with the chemical symbol A. Other molecules in equihbrium with A are A (n = 2,. . . ) with frictions O. In the actual solution, the fraction of the component which exists as A is written It is easily shown that the inverse of the component friction O is given by... 

The quantities KsOy Ks, and Kd are accessible through independent measurements and are independent of the molar mass. They are consequently called physical constants. On the other hand, A sd, A sr, and A dv are model constants, since they are based on certain assumptions. If, for example, the frictional coefficients from sedimentation and diffusion are of equal magnitude (see Chapter 9), XhttiAso = 1. The model constants can, of course, influence the numerical value of the molar mass, but they have no effect on the composition of the average from the various individual molecular species contributions. Consequently, model constants can always be assumed to have a value of unity until evidence to the contrary is obtained. 

The term ATm can be modeled following the Preston formalism (Rock et al., 2011) = PvKp, where P and v are the down pressure and the platen speed of CMP, respectively. ATp is an effective Preston coefficient. Kp = (mEm)/Bmc. with Bmc and Tm denoting the molar binding energy and molar volume of MC, respectively fi is the effective coefficient of friction of the pad—metal interface. In this description of chemically dominated CMP, one has MRR = 0 if the surface complex does not form. If no surface complexes or soluble species are formed, and if mechanical abrasion of the unmodified metal surface is the only means of material removal, then Eqn... 

Molar concentration or conductor or capacitance Gas concentration within agglomerate particle Concentration of species i at inlet of gas channel Skin friction Local skin friction Bulk concentration Gouy-Chapman capacitance Capacitance of the double layer... 

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