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The inescapable conclusion is that except in a minority of systems a step polymerization must be carried out as an open, driven system. That is, we must remove at least one of the products of the forward (polymerization) reaction so as to drive the equilibrium toward high molecular weights. It is usually more convenient to remove the small molecule byproduct rather than the polymer. When water is the by-product, it can be removed by a combination of temperature, reduced pressure, and purging with inert gas. Conveniently one often carries out step polymerizations at temperatures near or above the boiling point of water. This is usually done for purposes of obtaining desired reaction rates, but it has the added advantage of facilitating water removal. A small molecule by-product such as HCl can be removed in the same manner or by having a base present in the reaction system to neutralize the hydrogen chloride. Driving an equilibrium toward polymer requires considerable effort, since the water or hydrogen chloride or other small molecule must diffuse through and out of the reaction mixture. Diffusion is not so easy since the typical step polymerization system is fairly viscous at very high conversions. The polymerization can become diffusion-controlled under these conditions with the polymerization being controlled by the rate of diffusion of the small molecule by-product [Campbell et al., 1970]. The extent to which one must drive the system in the forward direction can be seen by calculating the lowering of the small molecule concentration, which is necessary to achieve a particular molecular weight. For the polyesteri cation (Eq. 2-58) one can rewrite Eq. 2-59 as



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which is analogous to the chain rule of ordinary calculus. This is all we are going to mention about stochastic calculus in this brief introduction to the subject. See some of the references in the recommended reading for a more comprehensive discussion of this vast field of mathematics.

K p H2 O M 0 1 p 2 2-63





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Now we can go back to our simple back-of-the-enveloperate equations for a laser and see the effect of noise. There are various sources of noise in a laser. There is noise from spontaneous emission, noise from the intrinsic quantum mechanical nature of light, pump noise, noise from the random decay processes that take the atoms or molecules from the lower energy level of the lasing transition to other levels that do not take part in the laser,15 and so on. Here we consider only spontaneous emission noise. This is often a good approximation [611].

8-6 8-5 8-4

8-1e-2

lSWhen chance leaves an atom/molecule in this lower level. it can absorb a photon from the cavity mode.

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As mentioned, you may find it easier to read and edit records in a form than in a large table containing other records. One way to locate records you want to view or edit is to use the various buttons in the navigation bar, such as the Previous Record button, the Next Record button and so on. (Refer to the preceding section for help using these buttons.) This method can become time-consuming, however, if the form contains many records. An easier approach is to search for the record using Access s search functionality, also accessible from the navigation bar.

The 1,4-polymerization of a 1-substituted 1,3-butadiene can yield four stereoregular polymer structures. The double bond can have cis or trans con gurations, and each of these can be

To describe spontaneous emission noise, we add a Langevin force F(t) to the rate equation for the photon number n. So (8.7) remains unchanged, but instead of (8.6), we now have (8.75) it = -Kn + /3"fnNn + Rs p + F, where

8-7

(F(t)F(t')} = D o(t - t').

combined with isotactic or syndotactic placement of R groups. (Other polymer structures, not referred to as stereoregular polymers, are possible with either random placement of double bonds or atactic placement of R groups.) The all-trans all-isotactic and all-trans all-syndiotactic structures for the 1,4-polymerization of 1,3-pentadiene are shown in Fig. 8-6. In naming polymers with both types of stereoisomerism, that due to cis trans isomerism is named rst unless it is indicated after the pre x poly. Thus, the all-trans all-isotactic polymer is named as transisotactic 1,4-poly(1,3-pentadiene) or isotactic poly(E-3-methylbut-1-ene-1,4-diyl). The sp3 stereocenter (i.e., C*) in XII is chirotopic, like the case of poly(propylene oxide), since the rst couple of atoms of the two chain segments are considerably different. The isotactic structures are optically active while the syndiotactic structures are not optically active. The 1,4-polymerization of 1,4-disubstituted 1,3-butadienes leads to structure (XIII), which can exhibit tritacticity since the repeating unit contains three sites of steric

(8.76)

8-8

What value should D have As we have seen in Section 8.2, D is the diffusion coefficient associated with the Brownian motion due to F. Thus, the variance of n is Dt. We can calculate this variance using a simple heuristic argument. As the only source of noise that we are considering here is spontaneous emission, if we separate from n the photons nsp that have been generated spontaneously, we will be able to treat the remaining number of photons as an ordinary (deterministic) variable. Let us write the average of the square of the photon number in the following way:

Fig. 8-6 Two of the stereoregular forms of 1,4-poly(1,3-pentadiene), [ CH2 CH CHCH(CH3 ) ]n .

= {n -

isomerism a double bond and the carbons holding R and R0 substituents. Several different stereoregular polymers are possible with various combinations of ordered arrangements at the three sites. For example, polymer XIV possesses an erythrodiisotactic arrangement of

+ nsp)(n - nsp + nsp)} nsp} {n - nsp} + 2 (n - nsp) (nsp) + (n~p)

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