Lauritzen - Hoffman Theory

The Lauritzen-Hoffman growth rate of polymer crystals is defined by  

Where is a pre-exponential factor containing quantities not strongly dependent on temperature, is a universal constant characteristic of the activation energy of chain motion (reptation) in the melt, R is the gas constant, T is the crystallization temp, T°o is the theoretical temperature at which all motion associated with viscous flow or reptation ceases, A is the nucleation parameter, AT = - T the undercooling, T is the equilibrium 

As indicated above, the term P allows for the diffusion of the stem to the site and is usually described by the WLF equation, h is Planck s constant, J is a dimensionless scaling constant, is a constant (dimensions J mol ) and is the temperature at which diffusion is stopped. If the polymer is of sufficiently high molar mass for entanglement to take place, then instead of the WLF theory the reptation theory of de Gennes should be used  

The ratios of AojB and AjB are independent of i//. The absolute rate constants are dependent on ij/ and the crystallization process involves subsequent addition of stems 1,2,. .., nto the substrate surface. If it is assumed that a steady-state condition is developed then  

The equations describe the process of sequential addition of stems to the substrate. The initial stems are deposited and described by A o o and a balance will be achieved between the fraction of the stems that are permanently attached to the crystal and those that return to the melt by the ratio of the backward to the forward process, i.e. BjA. 

This equation relates the rate of crystallization to its dependence on crystal thickness (L ) and temperature (Tc). The average crystal thickness can be 

The lower limit of the above equation corresponds to the case of a crystal formed at the melting point  

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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... 

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,...

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 2.26 Attachment of crystalline segments to a growing crystal according to the Lauritzen-Hoffman theory. Here g denotes the lateral growth rate and F the linear growth rate...

For this set of PE-h-PS diblock copolymers, several polymer nucleation and growth theories were applied to overall DSC isothermal crystallization rate data [61, 130,269,270]. These results indicate that the behavior of the PE-6-PS system is complex. Although the Lauritzen-Hoffman theory was developed for describing crystal growth only, it has been employed to describe overall crystallization data since it is capable of fitting the data remarkably well [269]. In these cases, such as when isothermal crystallization kinetics data obtained by DSC is employed, the energy barrier for crystallization reflected in Kg (denoted by us in this case KJ) contains contribution from both primary nucleation and crystal growth. 

Polyethylene-Based Blends, Composites and Nanocomposites 2.3.2 Lauritzen - Hoffman Theory... 

The effect of the blocky chain architecture on spherulite growth rate and bulk crystallization kinetics of novel ethylene-octene block copolymers is described. These copolymers form space-filling spherulites even when the crystdlinity is as low as 7 %. Spherulite growth rates were analyzed by Lauritzen-Hoffman theory and the bulk crystallization kinetics were subjected to Avrami analysis. Comparison with random copolymers showed that the blocky architecture imparts a substantially higher crystallization rate. 

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