Growth Front

If real growth fronts were to impinge on a point like this, their growth would terminate at x. Suppose we imagine point x to be charmed in some way such that any number of growth fronts can pass through it without interference. 

Our main interest here is concerned with patterns which can grow at constant speed even at low undercoolings A < 1, because if they exist they will dominate the system s behavior. A two-phase structure must then exist behind the growth front, filling the space uniformly on sufficiently... 

Fig. 4.2a, b. Configurations of molecules which are unfavourable to further growth, a The growth face of a lamella is shown, on which a molecule has deposited but is prevented from reaching the length required for stability by other attachments elsewhere, b Two examples of possible cross-sections perpendicular to the growth front. The outermost depositions must be removed before further growth of the stable crystal... 

The average thickness may easily be measured as the average stem length in the bulk part of the lamella, and the growth rate is the mean distance moved by the growth front during a number of Monte Carlo time-steps, nt, divided by nt. 

The average stem thickness is the average stem length at some position n, sufficiently far away from the growth front that the stem length is constant with n ... 

Interestingly, this barrier does not depend on chain length. This result coincides with experimental observations on the primary nucleation rate of bulk polymers [128-130]. For secondary nucleation of crystallization on a smooth growth front, a similar free-energy expression can be obtained for 2D nucleation ... 

Finally, we were led to the last stage of research where we treated the crystallization from the melt in multiple chain systems [22-24]. In most cases, we considered relatively short chains made of 100 beads they were designed to be mobile and slightly stiff to accelerate crystallization. We could then observe the steady-state growth of chain-folded lamellae, and we discussed the growth rate vs. crystallization temperature. We also examined the molecular trajectories at the growth front. In addition, we also studied the spontaneous formation of fiber structures from an oriented amorphous state [25]. In this chapter of the book, we review our researches, which have been performed over the last seven years. We want to emphasize the potential power of the molecular simulation in the studies of polymer crystallization. 

Real polymer processes involved in polymer crystallization are those at the crystal-melt or crystal-solution interfaces and inevitably 3D in nature. Before attacking our final target, the simulation of polymer crystallization from the melt, we studied crystallization of a single chain in a vacuum adsorption and folding at the growth front. The polymer molecule we considered was the same as described above a completely flexible chain composed of 500 or 1000 CH2 beads. We consider crystallization in a vacuum or in an extremely poor solvent condition. Here we took the detailed interaction between the chain molecule and the substrate atoms through Eqs. 8-10. 

Fig. 32 Changes in the lamella profiles viewed along the x-axis, during melting at 380 K a at 0.0 ns, b at 0.128 ns, c at 0.64 ns. We can see that the tapered growth fronts retreat at 380 K maintaining the tapered shape at the edges...

Fig. 37 Three dimensional pictures of growing lamellae of Ciooo at 370 K 3 at 0.128 ns, b at 6.4 ns, c at 12.8 ns, and d at 19.2 ns. We again see the tapered growth fronts and their advances in the normal (the z-axis) direction together with the lamella thickening along the chain axis (the y-axis direction)...

The natural questions that arise are what constitutes the stems, how the stems (if they exist) get attached at the growth front, what the free-energy barriers are, etc. The molecular modeling has provided vivid details for these questions, and has shown that the assumptions of the LH model are not valid in any universal way. The simulation results have clearly underscored the need for new theories of polymer crystallization by providing credence to the criticisms of the LH theory. 

In addition to visualizing the crystallization process, we also seek to determine the effect of varying the chain length relative to the lamellar thickness. Since the inclusion of chain ends inside the crystal is energetically unfavorable (akin to the inclusion of impurities), we can choose a case where the chain length is not an integer multiple of the lamellar thickness and see how the chain is accommodated on the growth front. 

For the first case, we have chosen a chain of length 100 at T = 8, which should fold once on a growth front of length 50. Figure 15 shows the sequence of one such event and the values of time are indicated in the frames. 

Fig. 15 Attachment of L = 100 chain onto L = 50 growth front model. Growth front chains are immobilized. The chain exhibits significant mobility (a-e) before it establishes perfect registration with the surface (f). The values of time t are indicated in each frame... 

---