Chemical potentials polydispersity

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. 

For a polydisperse polymer, analysis of sedimentation equilibrium data becomes complex, because the molecular weight distribution significantly affects the solute distribution. In 1970, Scholte [62] made a thermodynamic analysis of sedimentation equilibrium for polydisperse flexible polymer solutions on the basis of Flory and Huggins chemical potential equations. From a similar thermodynamic analysis for stiff polymer solutions with Eqs. (27) for IT and (28) for the polymer chemical potential, we can show that the right-hand side of Eq. (29) for the isotropic solution of a polydisperse polymer is given, in a good approximation, by Eq. (30) if M is replaced by Mw [41],... 

Here, /u ° and ju are, respectively, the chemical potentials of pure solvent and solvent at a certain concentration of biopolymer V is the molar volume of the solvent Mn=2 y/M/ is the number-averaged molar mass of the biopolymer (sum of products of mole fractions, x, and molar masses, M, over all the polymer constituent chains (/) as determined by the polymer polydispersity) (Tanford, 1961) A2, A3 and A4 are the second, third and fourth virial coefficients, respectively (in weight-scale units of cm mol g ), characterizing the two-body, three-body and four-body interactions amongst the biopolymer molecules/particles, respectively and C is the weight concentration (g ml-1) of the biopolymer. 

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]. 

The sedimentation process gives rise to a solvent phase and a concentrated polymer solution phase which are separated by a boundary layer in which the polymer concentration varies. There is, therefore, a natural tendency for backward diffusion of the molecules in order to equalise the chemical potentials of the components in the different regions of the cell, and this causes broadening of the boundary layer. The breadth of the boundary layer also increases with the degree of polydispersity because molecules of higher molar mass sediment at faster rates. The windows in the cell enable the radial variation in polymer concentration to be measured during ultracentrifugation typically... 

This development assumes that for a polydispersed polymer in a single solvent, all polymer components have the same specific volume and the same refractive index and that dn/dw is independent of the molecular weight distribution. Generally the chemical potentials determined from this technique are slightly higher than values determined by other methods. The difference, however, is believed to be within the experimental error. The method is applicable over a concentration range from 0 to 80 weight percent polymer. 

Xi increases slowly with concentration. It can be shown that if the chemical potential of a monomer in an aggregate is sharply distributed about some N, then the distribution of aggregates peaks at a value of N just less than the N with minimum fifp and is also sharply distributed. Otherwise, pronoimced polydispersity may occur e.g. for long cylindrical micelles). Thus reduced to bare bones it can be seen that the use of the word "theory" is dubious. We have simply characterised the observation of micelles, and claim that it can be shown that the law of mass action is an appropriate vehicle for this characterisation. 

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. 

The first term on the right-hand side characterizes the pair interaction between segments, the second term arises from the incompressibility constraint, and the last term contains the single-chain partition funaion Q[co] that depends on the potential fields nonlocal functional of the field variables and generally contains an imaginary part. The form of H ] co ] depends on the chain model used in the theory, on polymer architecture, chemical composition, polydispersity, and so on. 

As mentioned earlier, the aggregation number of micelles is not monodisperse but polydisperse. Therefore, their distribution is a matter of some concern. Let us take the model of micelle formation expressed by Eq. (4-1), where n is not definite but diffuse. Then, if /jl and fii are the chemical potentials of the micellar species composed of n monomers and the monomer, respectively, we have for the equilibrium between the monomers and any micellar species ... 

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... 

In closing this subsection, we note that the phase equilibrium calculation for polydisperse polymers is conceptually straightforward but mathematically tedious. Each polymer fi action has to be treated as a separate species with its own chemical potential given by an equation similar to Eq. (9.3.31). The interaction parameter, however, is taken to be independent of molecular weight. It is... 

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