Chemical potential, polydisperse system

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. 

Computationally, polydispersity is best handled within a grand canonical (GCE) or semi-grand canonical ensemble in which the density distribution p(a) is controlled by a conjugate chemical potential distribution p(cr). Use of such an ensemble is attractive because it allows p(a) to fluctuate as a whole, thereby sampling many different realizations of the disorder and hence reducing finite-size effects. Within such a framework, the case of variable polydispersity is considerably easier to tackle than fixed polydispersity The phase behavior is simply obtained as a function of the width of the prescribed p(cr) distribution. Perhaps for this reason, most simulation studies of phase behavior in polydisperse systems have focused on the variable case [90, 101-103]. 

Our purpose in this section is to derive a set of useful expressions for the chemical potentials starting with the principles of statistical mechanics. The expressions we shall obtain take the form of virial expansions similar to those of the Edmond and Ogston (6) but having a very different theoretical basis. Our model parameters are isobaric-isothermal virial coefficients which are about an order of magnitude smaller than the osmotic virial coefficients in the Edmond and Ogston model. We shall develop the theory neglecting the effect of polydispersity because we empirically did not find this to be very important at the level of accuracy commonly attainable in experimental phase diagrams for these systems. 

Thermodynamically, if there is an equilibrium between a solution and a solid state, the Gibbs free energies AG of a polymer species in solution and in the solid state are equal. The equilibrium concentration of the polymer species in solution is then called the saturated concentration, Cs. In the case of a polydisperse polymer, there will be polymer species with different MWs, both in solution and in the solid state, all of which will feature unique energies and saturated concentrations. To describe their contributions, chemical potentials of species /i, are used, which are partial molar quantities and represent the change in the overall Gibbs energy of the system upon addition of one mole of the species in question. 

In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. 

Higuchi and Misra were the first to show that if one of the components of a dispersed phase is completely insoluble in the continuous phase, then even small amounts of such a substance may stop the Ostwald ripening in the system. The reason for this is as follows. In a two-component dispersed phase system, the mass transfer of the more soluble component from small to larger drops caused by the difference in the Laplace pressures changes the composition of the drops. Namely, it increases the concentration of the poorly soluble component in the small drops and decreases it in the larger ones. According to Raoult s law, this results in a compensation of the difference in chemical potentials of the more soluble component caused by the difference in capillary pressures. When the capillary and concentration effects completely compensate, the mass transfer terminates and the drops come to equilibrium . This equilibrium implies the equality of the chemical potentials of the major component in all of the drops of the polydisperse emulsion. Such an equality is unattainable for the second component if its solubility in the continuous phase is truly zero. Kabalnov et have considered two cases as... 

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