Hoffman nucleation theory

For many polymers, the change between the initial lamellar thickness at T_c is roughly the same as at T_m and can thus be modeled by the Gibbs–Thomson equation fairly well. However, since it implies that the lamellar thickness over the given supercooling range (T_m–T_c) is unchanged, and many homogeneous nucleation of polymers implies a change of thickness at the growth front, Hoffman and Weeks pursued a more accurate representation.[4] In this regard, the Hoffman-Weeks plot was created and can be modeled through the equation

${\displaystyle T_{\text{m}}={T_{\text{c}} \over \beta }+(1-{1 \over \beta })T_{\text{m}}^{\circ }}$

where β is representative of a thickening factor given by L = L₀ β and T_cand T_m are the crystallization and melting temperatures, respectively.

Applying this experimentally for a constant β allows for the determination of the equilibrium melting temperature, T_m° at the intersection of T_cand T_m.[3]

The Lauritzen–Hoffman growth theory breaks the kinetics of polymer crystallization into ultimately two rates. The model breaks down into the addition of monomers onto a growing surface. This initial step is generally associated with the nucleation of the polymer. From there, the kinetics become the rate which the polymer grows on the surface, or the lateral growth rate, in comparison with the growth rate onto the polymer extending the chain, the secondary nucleation rate. These two rates can result in three situations.[6]

For Regime I, the growth rate on the front laterally, referred to as g, is the rate-determining step (RDS) and exceeds the secondary nucleation rate, i. In this instance of g >> i, monolayers are formed one at a time so that if the substrate has a length of L_p and thickness, b, the overall linear growth can be described through the equation

${\displaystyle G_{\text{I}}=biL_{\text{p}}}$

and the rate of nucleation in specific can further be described by

${\displaystyle G_{\text{I,n}}=e^{-(K_{\text{g}}/T\Delta T)}\,}$

with K_g equal to

${\displaystyle K_{\text{g}}={4b\sigma _{\text{l}}\sigma _{\text{f}}T_{m}^{0} \over k\Delta h}}$

where

• σ_l is the lateral/lamellae surface free energy per unit area
• σ_f is the fold surface free energy per unit area
• T_m⁰ is the equilibrium melting temperature
• k is equal to Boltzmann constant
• Δh is equal to the change of enthalpy of fusion (or latent heat of fusion) per repeat unit at the standard temperature[3]

This shows that in Region I, lateral nucleation along the front successfully dominates at temperatures close to the melting temperature, however at more extreme temperatures other forces such as diffusion can impact nucleation rates.

In Regime II, the lateral growth rate is either comparable or smaller than the nucleation rate g ≤ i, which causes secondary (or more) layers to form before the initial layer has been covered. This allows the linear growth rate to be modeled by

${\displaystyle G_{\text{II}}=b{\sqrt {ig}}}$

Using the assumption that g and i are independent of time, the rate at which new layers are formed can be approximated and the rate of nucleation in regime II can be expressed as

${\displaystyle G_{\text{II,n}}=e^{-(K_{\text{g}}^{\prime }/T\Delta T)}\,}$

with K_g^' equal to about 1/2 of the K_g from Regime I,

${\displaystyle K_{\text{g}}^{\prime }={2b\sigma _{\text{l}}\sigma _{\text{f}}T_{m}^{0} \over k\Delta h}}$

Lastly, Regime III in the L-H model depicts the scenario where lateral growth is inconsequential to the overall rate, since the nucleation of multiple sites causes i >> g. This means that the growth rate can be modeled by the same equation as Regime I,

${\displaystyle G_{\text{III}}=biL_{\text{p}}=G_{\text{III}}^{\circ }e^{{-U^{*}/k(T-T_{\text{0}})}-(K_{\text{g}}/T\Delta T)}}$

where G_III^° is the prefactor for Regime III and can be experimentally determined through applying the Lauritzen–Hoffman Plot.[7]

A reza's crystallization depends on the time it takes for layers of its chains to fold and orient themselves in the same direction. This time increases with a molecule's weight and branching.[8] The table below shows that the growth rate is higher for Sclair 14B.1 than Sclair 2907 (20%), where 2907 is less highly branched than 14B.1.[8] Here Gc is the crystal growth rate, or how quickly it orders itself depending on the layers, and t is the time it takes to order.