Molecular distribution most probable

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

Hence the sizes of spherical micelles are distributed around a most probable aggregation number M, which depends only on molecular details of the surfactants in this simplest approximation. Indeed, micelle size distributions at concentrations beyond the CMC have shown a marked peak at a given aggregation number in many simulations [37,111,112,117,119,138,144,154,157]. 

This distribution is known as the Schultz-Flory or most probable distribution.2S The moments of the molecular weight distribution are ... 

Because of the absence of chain limiter, the catalyst itself may initially act as the chain limiter (Fig. 8.22). The catalyst reacts with the olefinic regions of the polymer backbone and causes chain scission to occur, forming two new chains. The reactive carbene which is produced then moves from chain to chain, forming two new chains with each scission until the most probable molecular weight distribution is reached (Mw/Mn = 2), producing linear chains end capped with [Ru] catalyst residues. 

A pattern emerges when this molecular beam experiment is repeated for various gases at a common temperature Molecules with small masses move faster than those with large masses. Figure 5 shows this for H2, CH4, and CO2. Of these molecules, H2 has the smallest mass and CO2 the largest. The vertical line drawn for each gas shows the speed at which the distribution reaches its maximum height. More molecules have this speed than any other, so this is the most probable speed for molecules of that gas. The most probable speed for a molecule of hydrogen at 300 K is 1.57 X 10 m/s, which is 3.41 X 10 mi/hr. 

The viscosity average molecular weight depends on the nature of the intrinsic viscosity-molecular weight relationship in each particular case, as represented by the exponent a of the empirical relationship (52), or (55). However, it is not very sensitive to the value of a over the range of concern. For polymers having the most probable distribution to be discussed in the next chapter, it may be shown, for example, that... 

The other case which we consider is that of a most probable primary distribution. The molecular size distribution after random cross-linking must correspond exactly to that which would be obtained by random condensation of a mixture of bifunctional and tetrafunctional units. This follows as an extension of the correspondence between these two cases considered in the discussion of the critical condition given in the preceding section. The equations developed there are applicable to this case. 

The molecular weight distribution in Fig. 5.3 a) exhibits a most probable molecular weight distribution , which is characteristic of polymers produced by metallocene catalysts. This distribution contains relatively few molecules with either extremely high or low molecular weights. Products made with this type of distribution are relatively difficult to process in the molten state, exhibit modest orientation, and have good impact resistance. 

Resins with a bimodal molecular weight distribution, as illustrated in Fig. 5.3 b), are more readily processed than those with a most probable molecular weight distribution, but are more likely to be oriented in the solid state and have a somewhat reduced impact resistance. 

Resins with a significant low molecular weight tail, as shown in Fig. 5.3 c), exhibit similar melt flow characteristics to the most probable molecular weight distribution, but may be more flexible in the solid state due to the plasticizing effect of the shorter chains. 

In practice, product developers often blend two or more resins together in order to obtain a product that has the required melt flow and solid-state characteristics. Thus, we frequently combine metallocene catalyzed linear low density polyethylene, having a most probable molecular weight distribution, with low density polyethylene, having a broad molecular weight distribution. The linear low density polyethylene provides good impact resistance, while the low density polyethylene improves melt flow characteristics. 

The equations relating Mn and Mw to radiation dose which are most frequently used apply to all initial molecular weight distributions for Mn, but only to the most probable distribution (Mw/Mn = 2) for Mw. However, equations have been derived for other initial distributions, especially for representation by the Schulz-Zimm distribution equation. 

SOLUBLE/INSOLUBLE (GEL) FRACTION. If crosslinking predominates over scission (when G(crosslink) > 4 G(scission)), the decrease in soluble fraction above the gel dose, may be used to derive G values for both processes. An equation was derived by Charlesby and Pinner for the most probable molecular weight distribution and similar equations have been derived for other distributions. 

Different types of LCB are distinguished. Star polymers are the simplest branched polymers because they have only one branch point. Regular star polymers have a branch point with a constant number (functionality,/) of arms and every arm has the same molecular weight. They are therefore monodisperse polymers. Star polymers may also have arms with a most probable distribution [5], Star polymers can also be polydisperse due to a variable functionality. Palm tree [6] or umbrella polymers [7] that contain a single arm with different molecular weight (MW) than the other arms are classified under the asymmetric star [8] polymers, see Figure 3.2. 

The results of the various methods used to determine the molecular weight and degree of functionality of the diamine are shown in Table II. All of this evidence is consistent with the conclusion that the purified, isolated diamine possesses high difunctionality. When the diamine was chain extended with terephthaloyl chloride, a polymer product with a symmetrical molecular weight distribution was obtained (D.P. a 10). Inexact stoichiometry is most probably responsible for the limited extent of chain extension. 

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