Polydisperse system

If the form function of a monodisperse system is known, then the form function of a polydisperse system is [3] 

When interparticle correlations are involved, the calculation of the intensity function becomes more complex since it involves essentially all interactions between particle 1 with size Xi and particle 2 with size X2, particle 3 with size Xs, and so on. Fortunately, these systems can be approximated by the local monodisperse approximation (LMA), which assumes that particles are surrounded only by other particles of the same size [44]. Following this approximation, the corresponding scattering intensity can be described as 

In general, a solution contains molecules of different molecular weights, and in this case the scattering intensity is the sum of the contributions from the various molecules of molecular weight Mx with the concentration c or weight fraction wx = c,/c. The total concentration is c = For homopolymers, one now obtains 

Mw is the weight average molecular weight and Pz(q) the z-average of the particlescattering factor the particle-scattering factor of an x-mer in the ensemble is given by the expression in the brackets of Eq. (B.1S). 

For copolymers, the corresponding equation for the scattering intensity reads60,61) 

Here wxi denotes the weight fraction of an x-mer with a special composition and v is given by Eq. (B.14). The relationship Eq. (B.19) may conveniently be written as 

In the second part of Eq. (B.21 b), the refractive index increment of the i-th isomer has been used which is defined in a way similar to Eq. (B.14), i.e. 

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For preparative purposes batch fractionation is often employed. Although fractional crystallization may be included in a list of batch fractionation methods, we shall consider only those methods based on the phase separation of polymer solutions fractional precipitation and coacervate extraction. The general principles for these methods were presented in the last section. In this section we shall develop these ideas more fully with the objective of obtaining a more narrow distribution of molecular weights from a polydisperse system. Note that the final product of fractionation still contains a distribution of chain lengths however, the ratio M /M is smaller than for the unfractionated sample. 

For a polydisperse system containing molecules in different molecular weight categories which we index i, we can write (m,), =, and... 

For polydisperse systems the value of M obtained from the values of s° and D°-or, better yet, the value of the s/D ratio extrapolated to c = 0-is an average value. Different kinds of average are obtained, depending on the method used to define the average location of the boundary. The weight average is the type obtained in the usual analysis. 

This result uses the already established fact that M = when the molecular weight is determined by light scattering for a polydisperse system. 

Numerical results for the some model polydisperse systems have been reported in Refs. 81-83. It has been shown that the effect of increasing polydispersity on the number-number distribution function is that the structure decreases with increasing polydispersity. This pattern is common for the behavior of two- and three-dimensional polydisperse fluids [81] and also for three-dimensional quenched-annealed systems [83]. 

It would be interesting, but non-trivial, to study polydisperse systems by means of a simulation, both for distributions that allow large spheres and for distributions that prevent the existence of large spheres. The conceptual... 

The changes in the average chain length of a solution of semi-flexible selfassembling chains confined between two hard repulsive walls as the width of the sht T> is varied, have been studied [61] using two different Monte Carlo models for fast equihbration of the system, that of a shthering snake and of the independent monomer states. A polydisperse system of chain molecules in conditions of equilibrium polymerization, confined in a gap which is either closed (with fixed total density) or open and in contact with an external reservoir, has been considered. 

Platinum-cobalt alloy, enthalpy of formation, 144 Polarizability, of carbon, 75 of hydrogen molecule, 65, 75 and ionization potential data, 70 Polyamide, 181 Poly butadiene, 170, 181 Polydispersed systems, 183 Polyfunctional polymer, 178 Polymerization, of butadiene, 163 of solid acetaldehyde, 163 of vinyl monomers, 154 Polymers, star-shaped, 183 Polymethyl methacrylate, 180 Polystyrene, 172 Polystyril carbanions, 154 Potential barriers of internal rotation, 368, 374... 

In order to cadculate a particle size distribution directly from the output chromatogram for a polydisperse system, the integral, dispersion equation for the chromatogram signal, F(V), as a function of elution volume, V, needs to be evaluated (27) ... 

Equation (31) applies to monodisperse systems. For polydisperse systems Rg reflects a high-order moment of the distribution, the ratio of the 8th to the 6th moment of the distribution in mean size. For this reason Rg will correlate with the largest sizes of a distribution. There are several advantages to Rg as a measure of size over the end-to-end distance. For branched, star and ring structures the end-to-end distance has no clear meaning while Rg retains its meaning. Further, Rg is directly measured in static scattering measurements so it maintains a direct link to experiment. 

Beetstra, R., van der Hoef, M. A., and Kuipers, J. A. M. Numerical study of segregation using a new drag force correlation for polydisperse systems derived from lattice Boltzman simulations, Manuscript submitted the Chem. Eng. Sci. (2006, in press). 

Heterodisperse Suspensions. The rate laws given above apply to monodisperse colloids. In polydisperse systems the particle size and the distribution of particle sizes have pronounced effects on the kinetics of agglomeration (O Melia, 1978). For the various transport mechanisms (Brownian diffusion, fluid shear, and differential settling), the rates at which particles come into contact are given in Table 7.2. 

A numerical implementation of this approach can be generalised to include the polydispersity of the polymer. As polydispersity is increased the power law of — 0.8181... reduces. The onset of shear thinning, where Tn y 1 — 1, results in a slightly lower viscosity for polydisperse systems at this rate. So far we have neglected the origin of the characteristic time for the system which we would like to describe in terms of the chemical... 

Equation 2 describes the growth of each individual particle when its growth rate is determined by the diffusion of the solute from the bulk of the solution to the particle surface. The total consumption of solute from the solution may be obtained by expressing the growth rate as dn/dt = dV/dt = -4irr dr/dt, and adding the values of dn/dt for all the particles. This is simplest for a mono-disperse suspension, but may also be done for a polydisperse system. See Reference ( 3), page 301, and (34,35). ... 

HA are larger than FA and form polydisperse systems [43]. Precipitation is used to isolate HA from solids. The HA must aggregate to precipitate, and therefore the resulting polydisperse systems confirm that HA exist as aggregates of various sizes. HA are mixtures of a limited number of more or less chemically... 

Figure 21.3. Concentration distribution of solute in solution at sedimentation equilibrium. Curve A represents ideal behavior of a monodisperse solute curve B represents nonideality and curve C represents a polydisperse system.

A mathematical analysis of equilibrium behavior in a polydisperse system leads to the conclusion that from the slope at any point on curve C, we can obtain a weight average molecular weight at that local concentration of solute i. Several software programs are available for carrying out the necessary calculations [5]. 

In the case of a polydisperse system the calculation of the particle size distribution is possible by using special transformation algorithms. For this purpose certain requirements need to be fulfilled, such as a spherical particle shape, sufficient dilution, and a large difference between the refractive indices of the inner and the outer phases. Since usually not all requirements can be fulfilled, the z-average is preferred as a directly accessible parameter rather than the distribution fimction depending on models. 

It is probable that numerous interfacial parameters are involved (surface tension, spontaneous curvature, Gibbs elasticity, surface forces) and differ from one system to the other, according the nature of the surfactants and of the dispersed phase. Only systematic measurements of > will allow going beyond empirics. Besides the numerous fundamental questions, it is also necessary to measure practical reason, which is predicting the emulsion lifetime. This remains a serious challenge for anyone working in the field of emulsions because of the polydisperse and complex evolution of the droplet size distribution. Finally, it is clear that the mean-field approaches adopted to measure > are acceptable as long as the droplet polydispersity remains quite low (P < 50%) and that more elaborate models are required for very polydisperse systems to account for the spatial fiuctuations in the droplet distribution. 

Another very important physical parameter one must consider is the size distribution of the colloids. A system consisting of particles of the same size is called a monodis-perse. A system with different sizes is called polydisperse. It is also obvious that systems with monodisperse will exhibit different properties from those of polydispersed systems. In many industrial application (such as coating on tapes used for recording music and coatings on CDs or DVDs), latter kind of quality of coatings is needed. 

In the past few decades, a specific kind of colloidal system based on monodis-perse size has been developed for various industrial applications. A variety of metal oxides and hydroxides and polymer lattices have been produced. Monodisperse systems are obviously preferred since their properties can be easily predicted. On the other hand, polydisperse systems will exhibit varying characteristics, depending on the degree of polydispersity. 

Holland, A. C., and G. Gagne, 1970. The scattering of polarized light by polydisperse systems of irregular particles, Appl. Opt., 9, 1113-1121. 

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