Hoffman equation

With = const we get immediately the famous Hoffman-Weeks equation (Hoffman and Weeks, 1962a Hoffman and Weeks, 1962b) ... 

Gibbs-Thomson equation Hoffman-Weeks approach Isothermal crystallization Non-isothermal crystallization Rates of crystallization... 

Intrinsic stress cr at a temperatme of interest in the coinse of gel-to-ceramic thin film conversion can be obtained when the radii of the cmvature of the bare (R ) and coated substrate (il2) are known as well as the biaxial elastic modulus of the substrate (Es/(1 — v )) and the thickness of the substrate (fs) and film (ff), using Stoney s equation (Hoffman, 1966) ... 

Kouri D J, Huang Y, Zhu W and Hoffman D K 1994 Variational principles for the time-independent wave-packet-Schrddinger and wave-packet-Lippmann-Schwinger equations J. Chem. Phys. 100... 

To account for different strengths in tension and compression, Hoffman added linear terms to Hill s equation (the basis for the Tsai-Hill criterion) [2-23] ... 

Increased curve-fitting capability over the Tsai-Hili and Hoffman criteria because of an additionai term in the equation. The additionai term, F 2, can be determined only with an expensive and difficuit-to-perform biaxial test. 

The thermal Diels-Alder reactions of anthracene with electron-poor olefinic acceptors such as tetracyanoethylene, maleic anhydride, maleimides, etc. have been studied extensively. It is noteworthy that these reactions are often accelerated in the presence of light. Since photoinduced [4 + 2] cycloadditions are symmetry-forbidden according to the Woodward-Hoffman rules, an electron-transfer mechanism has been suggested to reconcile experiment and theory.212 For example, photocycloaddition of anthracene to maleic anhydride and various maleimides occurs in high yield (> 90%) under conditions in which the thermal reaction is completely suppressed (equation 75). 

Industrial synthesis of vitamin A (Hoffman-La-Roche) goes through partial hydrogenation of an enyne (equation 161)277. A number of syntheses of pheromones, where the reduction of an enyne to a diene is the key step, have been devised. A few selected examples are given in Table 29278. During the total synthesis of endiandric acids, Nico-laou employed hydrogenation of a polyenyne intermediate with a Lindlar catalyst to generate an intermediate which underwent symmetry-allowed cyclizations to result in the natural product (equation 162)279. 

As reported by Hoffman and Buntain cyclic A-((4-nitrophenyl)sulfonoxy)amines (587) rearrange at low temperatures to ring-expanded cyclic imines 588 in fair to good yields (42-82%) (equation 259). A competitive migration of the 1,2 and 2,3 bonds is observed and an ionic pathway is proposed. 

Hoffman and coworkers have extensively studied the reactions between amines and sulfonyl peroxides. When primary amines reacted with arylsulfonyl peroxides at —78 °C in ethyl acetate, A-alkyl-O-arylsulfonyl hydroxylamine derivatives were obtained (equation 8), whereas when various primary and secondary amines reacted with sulfonyl peroxides, oxidative deamination was observed. ... 

The lamellar habit adopted by crystalline polymers adds surface terms to the specific Gibbs function (chemical potential), most importantly the fold surface free energy, ae, which contributes 2ae/Xg for a lamella of thickness k and crystalline density q. In consequence melting points are lowered from T, for infinite thickness, to Tm according to the Hoffman-Weeks equation... 

This equation has been widely used although the validity of applying the WLF equation to spherulitic growth rate is merely a repetitive assertion (Hoffman et al., 1959 Hoffman and Weeks, 1962), not involving any direct proof of substantiation, as Mandelkern has stated. 

Nitrogen-13 labelled / -phenethylamine [13N]-309 has been synthesized294 by Hoffman rearrangement of [13N]phenylpropionamide, prepared in turn from phenylpropio-nyl chloride and aqueous [13N]ammonia solution (equation 127). 

P 0 corresponds to the well-mixed reactor, P to plug flow. Solutions of Equation (12) for a selected set of positive finite values of P fills in the region between the two idealized reactor types. Hlavacek and Hoffman performed the necessary numerical calculations for a few selected values of the parameters and presented their results in a set of plots. Since their plots were intended for illustration rather than for quantatative use in design, I have repeated the calculations for a selected set of parameters. 

A very simple approximate solution to the above equations can also be obtained by a linearization technique suggested by Hlavacek and Hoffman.16 For example, Eqs. (4-73) through (4-77) can be rewritten as... 

The theory of Lauritzen and Hoffman, perhaps still the most commonly used model for the analysis of polymer crystallization data, then seeks to evaluate c5Z (usually of the order of 40 angstroms) by considering the rates at which stems and folds are successively laid down. We will not go into the details of this derivation, but the expressions that have been derived for the growth rate have the following form at low undercoolings (Equation 10-41) ... 

Hoffman, R., and W. N. Lipscomb (1962). Theory of polyhedral molecules. I. Physical factorization of the secular equation. II. A crystal field model. J. Chem. Phys. 36, 2179-95. 

Figure 18.8 Analysis of crystallization rate of SPS according to Lauritzen-Hoffman nucleation theory [Equation (1)] (AT = 7, -7), (a) Data plotted for various Mw O 4.70 X 104 1.26 x 105 tx, 3.78 x 105 A, 6.91 x 105 o, 9.23 x 105 , 2.00 x106. (b) Superposed line by horizontal shift with reference to 7°/TAT...

Bandermann and coworkers studied the MMA polymerization initiated by 37 . Whenever the initiator is contaminated by traces of diethyl 2-ethylmalonate, an inhibition period is observed, which was accounted for by a transfer reaction of the propagating enolate to diethyl 2-ethylmalonate. The polymerization cannot start until this contaminant is completely consumed. Consistently, no inhibition is observed when the initiator is highly pure. Finally, a Hoffman elimination in the propagating ammonium enolate (38) could occur with formation of dead chains (39) tri-n-butylamine and 1-butene (equation 43) . 

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