Effect of polydispersity

This leads to an expression relating the MWD to the viscosity average molecular weight M that was introduced in Eq. 2.45. 

In terms of this average, the MHS equation for a polydisperse sample is  

We note that depends strongly on a. This parameter varies from 0.5 to 1, and this implies that Mjj My M , so that My = M corresponds to a = 1. For theta conditions, a = 0.5, but for many common polymer-solvent pairs, a is closer to 0.8, so that the viscosity average molecular weight is usually closer to than to M . 

The sum in Eq. 2.96 can be replaced by an integral for use with a continuous molecular weight distribution to give  

One can then derive the relationship between the intrinsic viscosity and various molecular weight averages, if a distribution function is specified. For example, for the Schulz-Zimm distribution function (Eq. 2.61), we obtain the following relationships between [rj] and the weight and number average molecular weights. 

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Muthukumar [44] further investigated the effects of polydispersity, which are important for crosslinking systems. He used a hyperscaling relation from percolation theory to obtain his results. If the excluded volume is not screened, n is related to df by... 

Polydispersity is one of the most frequent reasons that soft condensed matter does not show diffraction but scattering. Thus its consideration is of utmost importance. The general effect of polydispersity on scattering patterns is demonstrated in this section. 

There are two important issues concerning the factor that gives the excluded volume 1 /shear rate (ii) what is the effect of polydispersity ... 

Fig. 3.6. Effect of polydispersity on the relation between the number averages of reduced stored free energy and reduced shear rate for a series of polystyrenes (Table 3.3) in monobromo-benzene at 25° C (75). The following concentrations given in g/100cms are used PS III () ) 3.0, (4) 1.5, (a) 0.75 DII (o-) 0.80, (-o) 0.575, () 0.39 DIV ( -) 1.0, ( ) 0.75, (- ) 0.55 F (>j) 1.0, (A) 0.7, ( ) 0.4 Taps. No. 15 ( -) 1.0, (- ) 0.633, (f ) 0.4. The full line indicates the non-draining behaviour of a monodisperse polymer (80). The dotted line is the result of extrapolating the data for PS III to zero concentration. The (MJMn)-ratios of the samples are given near the experimental points...

As a result of polydispersity effects, the composition of the incipient 13-phase segregated at the cloud point is located on a shadow curve, outside the cloud-point curve (point (3 in Fig. 8.4). The effects of polydispersity on phase diagrams and phase compositions may be found in specialized reviews (Tompa, 1956 Kamide, 1990 Williams et al., 1997). Because < )Mo < ( M,crit(xcp), the incipient (3-phase, which is richer in the modifier, will be dispersed in the a-phase, which is richer in the growing thermosetting polymer. The opposite occurs when < )M0 > M,crit(xcp)- It has been shown both theoretically (Riccardi et al., 1994 and 1996 Williams et al., 1997), and experimentally (Bonnet et al., 1999) that... 

Pronk, S. Frenkel, D. Large effect of polydispersity on defect concentrations in colloidal crystals. J. Chem. Phys. 120, 6764-6768 (2004). 

G. Marrucci and N. Grizzuti, Predicted effect of polydispersity on rodlike polymer behaviour in concentrated solutions, J. Non-Newt. Fluid Mech., 14,103 (1984). 

A.W. Chow, and G.G. Fuller, The rheo-optical response of rod-like chains subject to transient shear flow. Part I Model calculations on the effects of polydispersity, Macromolecules 18, 786 (1985) A.W. Chow, G.G. Fuller, D.G. Wallace and J.A. Madri, The rheo-optical response of rod-like chains subject to transient shear flow. Part II. Two-color flow birefringence measurements, Macromolecules 18,793 (1985) A.W. Chow, G.G. Fuller, D.G. Wallace and J.A. Madri, The rheo-optical response of rod-like shortened collagen protein to transient shear flow, Macromolecules, 18, 805 (1985). 

G. Murrucci and N. Grizzuti, The effect of polydispersity on rotational diffusivity and shear viscosity of rodlike polymer in concentrated solutions, J. Polym. Sci., Polym. Lett. Ed., 21, 83 (1983). 

Effect of Polydispersity of the Particle Size on the Depletion Force... 

Recently, Chu et al. [41,42] and Chu and Wasan [43] have made simulation and MSA studies of the structure and forces in thin films and colloidal dispersions. They include the effect of polydispersity. They also find oscillatory forces. 

To study charging mechanisms theoretically for either diffusion charging or field charging, it is necessary to make several assumptions regarding the aerosol. First, the particles are assumed to be spherical. This assumption is reasonable for isometric particles. Second, it is also assumed that the particles are monodisperse. The effect of polydispersity complicates but does not invalidate theory. Third, there are no interactions between individual particles. Finally, the ion concentration and electric field near each particle are assumed to be uniform. These last two assumptions are essentially true for all natural and industrial aerosols. Thus except in the most extreme cases, theoiy should be adequate without other modification. 

Jiang Y, Chen T et al (2005) Effect of polydispersity on the formation of vesicles from amphiphilic diblock copolymers. Macromolecules 38 6710-6717... 

Bushell, G., Amal, R., and Raper, J., The effect of polydispersity in primary particle size on measurement of the fractal dimension of aggregates. Part. Syst. Charact., 15, 3, 1998. 

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. 

We hope to extend this approach to other polymer systems and to include the effect of polydispersity in a forthcoming paper (Cabezas, H., Jr. Evans, J.D. Szlag, D.C. Huid Phase Equilibria, in press). 

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