Equilibrium lamellar thickness

Motivated by our simulation results, we now consider a theoretical model [32] which allows an exact calculation of the equilibrium lamellar thickness. Consider a nucleus sketched in Fig. 11, of thickness L = ml and radius R. For each of the n chains, let there be // stems (and // - 1 loops and two chain tails), each of length L. 

Having discussed some equilibrium properties of a crystal, we now outline and contrast the bases of the growth theories which will be dealt with in more detail later. The theories may be broadly split into two categories equilibrium and kinetic. The former [36-42] explain some features of the lamellar thickness, however the intrinsic folding habit is not accounted for. Therefore, at best, the theory must be considered to be incomplete, and today is usually completely ignored. We give a brief summary of the approach and refer the interested reader to the original articles. The kinetic theories will be the topic of the remainder of the review. 

At this point a third intermediate approach deserves mentioning. It is due to Allegra [43] who proposed that polymer crystallization is controlled by a metastable equilibrium distribution of intramolecular clusters, the so-called bundles , forming in the liquid phase. These subsequently aggregate to the side surfaces of the crystals, driven by van der Waals interactions. The lamellar thickness is determined by the average contour length of the loops within the bundles. Although the model can... 

A further paper [167] explains the lamellar thickness selection in the row model. The minimum thickness lmin is derived from the similation and found to be consistent with equilibrium results. The thickness deviation 81 = l — lmin is approximately constant with /. It is established that the model fulfills the criteria of a kinetic theory Firstly, a driving force term (proportional to 81) and a barrier term (proportional to /) are indentified. Secondly, the competition between the two terms leads to a maximum in growth rate (see Fig. 2.4) which is located at the average thickness l obtained by simulation. Further, the role of fluctuations becomes apparent when the dependence on the interaction energy e is investigated. Whereas downwards (i.e. decreasing l) fluctuations are approximately independent... 

A very recent application of the two-dimensional model has been to the crystallization of a random copolymer [171]. The units trying to attach to the growth face are either crystallizable A s or non-crystallizable B s with a Poisson probability based on the comonomer concentration in the melt. This means that the on rate becomes thickness dependent with the effect of a depletion of crystallizable material with increasing thickness. This leads to a maximum lamellar thickness and further to a melting point depression much larger than that obtained by the Flory [172] equilibrium treatment. 

The lamellar thickening proceeds through many metastable states, each metastable state corresponding to a particular number of folds per chain, as illustrated in Fig. 8. In the original simulations of [22], Kg was monitored. Rg is actually very close to the lamellar thickness due to the asymmetric shape of the lamella. The number of folds indicated in Fig. 8 were identified by inspection of the coordinates of the united atoms. This quantization of the number of folds has been observed in experiments [50], as already mentioned. The process by which a state with p folds changes into a state with p - 1 folds is highly cooperative. The precursor lives in a quiescent state for a substantial time and suddenly it converts into the next state. By a succession of such processes, crystals thicken. If the simulation is run for a reasonably long time, the lamella settles down to the equilibrium number of folds per chain. 

Independent of crystallization conditions, whether from solutions or melt, the polymer molecules crystallize into thin lamellae. The lamellar thickness is about 10 nm, about two orders of magnitude smaller than values allowed by existing equilibrium considerations. This is in contrast to the case of crystallized short alkanes, where the lamellar thickness is proportional to the length of the molecules. Clearly the chains in the case of polymers should fold back and forth in the lamellae to support the experimentally observed lamellar thickness. It is believed in the literature [3-9] that the lamellar thickness is kinetically selected and that if enough time is permissible, the lamella would thicken to extended chain crystal dimension. What determines the spontaneous selection of lamellar thickness ... 

The estimated value of the free energy of the fold surface (q3 = oy) is 90 mJ/m for polyethylene, whereas that of the lateral surface (ai = a/) is 15 mJ/m. Therefore we expect the lamellar thickness to be 6 times larger than the lateral dimension, specifically, a cylinder shape, instead of a disklike shape. This is in stark contrast to the facts described in Section II. The thermodynamic estimate of lamellar thickness is about two orders of magnitude larger than the observed values for polyethylene and other polymers. In view of this discrepancy, we are led to the conclusion that lamellae are not in equilibrium. 

The minimum in F(L) is observed to be near L/ q — 9 for aU chain lengths examined. It is to be noted that this minimum is the global minimum and the barrier between this state and other thicker lamellae increases prohibitively as the thickness increases. These simulations strongly suggest that a lamellar thickness that is much smaller than the extended chain thickness is actually an equilibrium result. 

The scaling of the lamellar thickness with degree of polymerization of the crystalline and amorphous blocks was investigated for PEO-poly(ferf-butyl methacrylate) (PEO-PtBMA) diblocks using DSC and SAXS by Unger et al. (1991). The non-equilibrium exponents obtained immediately after bulk crystallization were found to be different to those from equilibrium results extrapolated... 

This difference in behavior must reflect the difference in energetic barrier in the different cases, perhaps due to differences in the degree of thickening required in the transition between the different forms, the proximity to the equilibrium melting point and also perhaps to differences in the initial lamellar thickness. 

The tendency to form lamellar structures can be understood as the result of dipole-dipole repulsions between the uniaxially ordered amphiphile molecules theoretical analyses of the pattern formation based on this principle have been carried out. It appears that the lamellar thickness is defined by the equilibrium thermodynamic conditions (Fig. 23) but that the shape and density of the patterns are determined by the dynamics of the process by which they are formed. 

For polymers manifesting the most common type of crystalline morphology (folded chain lamellae), the "equilibrium" values (asymptotic limits at infinite lamellar thickness) of Tm, of the heat of fusion per unit volume, and of the surface free energy of the lamellar folds, are all lowered relative to the homopolymer with increasing defect incorporation in the crystallites. By contrast, if chain defects are excluded completely from the lamellae, the equilibrium limits remain unchanged since the lamellae remain those of the homopolymer, but the values of these properties still decrease for actual specimens since the average lamella becomes thinner because of the interruption of crystallization by non-crystallizable defects along the chains. 

Plotting the Tin obtained by DSC versus the inverse of the lamellar thickness (1/4), a straight line was observed. From the intercept, the equilibrium melting temperature... 

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