Monodisperse distribution

For the transformation of the macrocomposite model to a molecular composite model for the ultimate strength of the fibre the following assumptions are made (1) the rods in the macrocomposite are replaced by the parallel-oriented polymer chains or by larger entities like bundles of chains forming fibrils and (2) the function of the matrix in the composite, in particular the rod-matrix interface, is taken over by the intermolecular bonds between the chains or fibrils. In order to evaluate the effect of the chain length distribution on the ultimate strength the monodisperse distribution, the Flory distribution, the half-Gauss and the uniform distribution are considered. 

The most simple chain lenght distribution is the delta function or the monodisperse distribution w)=<5(w-wa). All chain lengths are equal to wa.The maximum of S is reached just prior to debonding, i.e. for wc=wa... 

Before we discuss the relation between ultimate fibre strength and degree of polymerisation (zn or DP), we first show that this model in fact describes the relation between the composite strength and the aspect ratio of the rod. In the case of a monodisperse distribution, i.e. all rods or chains have the same diameter 2r and the same length a, the ultimate strength as a function of the aspect ratio fr= a(2r) 1 is given by... 

In order to investigate, for a monodisperse distribution of the chain length, the effect of the degree of polymerisation on the fibre strength we write Eq. 89 as... 

Fig. 42 Ultimate strength aL of PpPTA fibre as a function of the degree of polymerisation for a monodisperse distribution calculated for series of diameters 2r of the building element, ju=0.16 and for aspect ratios b=ua(2r) 1>10...

Fig. 43 Double logarithmic graph of the ultimate strength of PpPTA fibre versus the degree of polymerisation for a monodisperse distribution for various values of the diameter and for /j=0.16. Calculation for aspect ratios fr=wa(2r) 1>10... 

For the same values of ea g, r0 and u0 as for the monodisperse distribution, Figs. 45 and 46 show the results for the Flory distribution of chain lengths. The curves start at degrees of polymerisation determined by zn=[(2a)-1+l]. A comparison of Fig. 45 with Fig. 42 shows that, for a diameter equal to the chain di-... 

As stated above (see Chapter II.B), LII signals also contain information about the size distribution. To compare the influence of different plasma powers on primary particle diameters, different ways of size evaluation have been accomplished. It could be shown by assuming a monodisperse distribution that the mean primary particle diameter is 31 nm for 30 kW and 33 nm for 70 kW. In contrast, under the assumption of a log-normal distribution and by applying the two-decay time evaluation, the determination yields a different result which can be seen in Figure 15. Size distributions with median sizes of 17nm and 28 nm and standard deviations of 0.39 and 0.18 for 30 kW and 70 kW were observed, respectively. This indicates that in practical production systems, the evaluation of a mondisperse distribution is not sufficient. Unfortunately, the reconstruction of particle size distributions is relatively sensitive on... 

Narrow particle fractions approaching a monodisperse distribution are particularly easy to treat and characterize when the above equations are applied to experimental data. Figure 2 shows an example of the elution profile (fractogram) obtained by running a mixture of four samples of "monodisperse" polystyrene latex beads. It is clear from the figure that a rather precise value of retention volume Vr can be identified with each bead size. With Vr known, it is easy to obtain R and X from Equation 5 and thence particle diameter d from Equation 4. This operation, as noted, yields diameters accurate to approximately 1-3%. 

The free energy of surfactant interfaces is due to interactions between water and the surfactant head-groups, as well as interactions between the surfactant chains, both of which compete to set the curvatures of the interface. Consequently, all else being equal, homogeneous interfaces are preferred over other geometries for a monodisperse distribution of surfactant... 

Polymer microspheres with monodispersed distribution in size are much perferable for phagocytosis assays, because the size dominantly governs phagocytosis of the microspheres by M< ). In addition, the microspheres should be prepared without any surfactants to exclude the influence of the soap molecules adsorbed onto the surface, on phagocytosis of the microspheres. We have synthesized monodispersed polystyrene microspheres by soap-free emulsion polymerization of styrene at 70 °C for 30 hr using potassium persulfate as a initiator [14]. The widely different diameters of the... 

The polydispersity of a sample is described by its molar mass distribution. Polydisperse and monodisperse distributions are sketched in Fig. 1.17. A distribution is shown as n, the number fraction (or mole fraction) of molecules containing N monomers each, plotted as a function of molar mass — of the molecules. 

Metal and semiconductor nanoparticles can be prepared in sizes ranging from a few nanometers to 0.1 pm or larger. Solution (wet chemical) preparation methods are available to produce nanoparticles with fairly monodisperse distributions and sizes that are unreachable with lithographic... 

Fig. 12. End-group initiation, initial monodisperse distribution [19]. The effect of chain transfer on the relative rate of weight loss, (6Mt ldt)l (dM,/df)0. Curves are plotted at one value of the initial zip length, 1/7° = 5 [(1/7°)/ ° = 0.0079], for several values of the transfer parameter times initial d.p., a°x° = (ktRJk )x0.

As earlier, consider ti to be the characteristic coagulation time of a polydisperse ensemble of drops, caused by the mechanism of turbulent diffusion due to the forces of hydrodynamic and molecular interactions. This time should be estimated. For typical values of the flow, Pq = 40 kg m , 2o = 5 x 10 m, Pq = 1.2 X 10 Pa-s, W = 5 X 10 m /m and distribution parameters of = 10 m, k = 3, one obtains 1/ti = 0.257 s. Thus, a twofold increase in drop radius occurs in a time t of 7 s. This time is almost two orders of magnitude higher than for a monodisperse distribution without regard to hydrodynamic and molecular forces. Such a big difference in characteristic times is undoubtedly caused not by taking into account the polydispersivity of the distribution, but as a result of considering the interaction forces. 

The solution obtained corresponds to a monodisperse distribution of drops without regard for coagulation. We consider a possible solution taking into account a polydisperse distribution and the coagulation of drops. 

The dependence of rj on dimensionless time r = t/tav is shown in Fig. 23.12. Note that k = oo corresponds to a monodisperse distribution of bubbles over volumes. 

In this chapter, all the results of nearly monodisperse systems presented show that the linear-additivity rule is well followed, except mainly for one sample, whose molecular weight is the highest among the studied. The deviation from the linear-additivity rule for this particular sample is small yet clearly detectable. The deviation was found to be due to the presence of a low-molecular-weight tail in the otherwise perfectly nearly monodisperse distribution of the sample. Because this effect has some bearing on the actual approach taken to analyze the G t) line shapes of the less perfect samples, we briefly discuss it below. 

As shown above, the G t) line shapes of a series of polystyrene samples at different molecular weights above M are well described by convolut-ing Eq. (9.19) with a nearly monodisperse distribution. Furthermore, the frictional factor K obtained from the analysis of the G t) curves measmed at the same temperature is independent of molecular weight. Thus, we expect to obtain good agreement between theory and experiment for the... 

For a monodisperse distribution of pore radii, this result is in good agreement with the values obtained fi om pore size distribution measurements, but it can be significantly in error if one is dealing with a bimodal pore size distribution (see Section 6.4.2). 

A better criterion in defining a microemulsion could be its fluidity. In effect, the viscosity of (macro)emulsions and suspensions is known to increase as the fragment size decreases, and thus it is expected that an emulsion with extremely small drop size, an internal phase content greater than 20-30%, and a monodispersed distribution (as expected in a microemulsion) would be quite viscous. However, systems with such a high viscosity have been called gel emulsions or miniemulsions because the authors preferred to elude the label microemulsion in order to avoid confixsion with single-phase microemulsions [5-7]. 

Assuming a monodisperse distribution of spherical water droplets, the average radius of the inner water cores is given by... 

In Fig. 30 theoretical P,(N) curves are shown when is variable and = const, b = const. As follows from the data of this figure, the value exerts a primary influence on the width of MWD. Strictly monodisperse distribution can be obtained only at = 0. The increase of the stochastic contribution in polycondensation intensity results to symmetrical broadening of MWD relative to its maximum. 

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