Hoffman theory

in Treatise on Solid State Chemistry, N. B. Han-nay, editor, Pergamon Press, Oxford (1989). 

This is half the value determined for primary nucleation. But, as in primary nucle-ation, the fold period goes as 1/tsT. In other words, it increases with decreasing undercooling (smaller AT). The temperature dependence of the rate of secondary nucleation has a different dependence on temperature, however. 

An expression for the temperature dependence of the rate of secondary nucleation can be obtained in a similar maimer to the procedure used in the treatment of primary nucleation. The expression for Vmin is substituted into the expression for the free energy, in this case the free energy of laying down a stem, AGacm, to obtain AG the free energy of a secondary nucleus that has the critical size. The rate of secondary nucleation can then be obtained from Equation 10-37  

We ll come back to this in a while. First let s look at the overall rate of crystallization, the quantity that can be experimentally determined. 

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The premise that nucleation was always the rate controlling factor in kinetic theories was first disputed by Sadler in 1983 [44]. The disagreement arises from a comparison of the morphologies which would be obtained using the free energies from the Lauritzen-Hoffman theory with those observed experimentally. The... 

In the classical Lauritzen-Hoffman theory for the mechanism of polymer crystal growth [106], it is assumed that the observed lamellar thickness corresponds to those crystallites that happen to have the largest growth velocity. However, this picture is hard to reconcile with the experimental observation that the thickness of polyethylene single crystals can be modulated by varying the temperature at which they are grown [117,118]. In fact, simulations by Doye et al. [119,120] suggest that the observed lamellar thickness does... 

Woodword-Hoffman theory states that there is high critical energy for the 1,3-sigmatropic shifts. However, theoretical calculations [28] and experimental data... 

Fig. 8 a Spherulitic growth rates for PPDX and the PPDX block within D7732C2310 diblock copolymer. Solid lines are fits to Lauritzen and Hoffman theory, b Lauritzen and Hoffman kinetics theory plot for PPDX (K = 17.2 x 104 K2) and the PPDX block within D7732C2310 diblock copolymer (K = 46 x 104 K2). (From [103]. Reproduced with permission of the Royal Society of Chemistry)... 

V. GROWTH, SECONDARY NUCLEATION, AND LAURITZEN-HOFFMAN THEORY... 

Fig. 5.34 Schematic of the growth rate in the three regimes of the Lauritzen Hoffman theory. Here AT = 7 J, - T where J 2, is the equilibrium melting point and Tc is the crystallization temperature,...

Lauritzen, J. I., jr., and J. D. Hoffman Theory of formation of polymer crystals with folded chains in dilute solutions. J. Research Natl. Bur. Standards 64 A, 73-102 (1960). 

Figure 50. Free energy of a new layer of stems on the lateral growth face of a polymer crystal as a function of stem length 1 and the number of stems in the layer, j. Calculated according to the Lauritzen-Hoffman theory, with p = 1. (a) AT= 11 °C, (b) AT = 80 °C. Standard data for polyethylene were used. Case a predicts 1 = lmin + 61 with a small 61, while (b) predicts 7— ...

It is assumed in the Lauritzen-Hoffman theory that, once a chain segment has been laid down on the growing surface parallel to the chains already forming the crystal growth face, the next units in the chain continue to be laid down to form a straight stem. When a lamellar face is reached the chain folds and the following units are laid down adjacent to... 

Fig. 5.19 Addition of chain segments to a growing crystallite according to (a) the Lauritzen-Hoffman theory and (b) the Sadler-Gilmer theory. Note that, in the Sadler-Gilmer theory, there are no perfectly regular fold surfaces, ((a) Adapted by permission of Kluwer Academic Publishers.)...

Figure 6.3 shows how different concentrations of radiation vulcanized polybutadiene rubber influences nucleation constant of polypropylene. The use of polybutadiene rubber promoted the crystals to become perfect and resulted in a larger crystallite size, which caused nucleation constant to decrease in value. This is compatible with Hoffman theory which predicts lower value of nucleation constant for neat PP because of heterogeneous nucleation effect of addition of vulcanized polybutadiene robber. Figure 6.4 shows the effect of exfoliated graphite on nucleation constant of PP com-... 

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