Stability of the sectored morphology of polymer crystallites

When an entangled interpenetrating collection of long flexible polymer chains dispersed in a suitable solvent is cooled to low enough temperatures, thin lamellar crystals form. Remarkably, these lamellae are sectored, with several growth sectors that have differing melting temperatures and growth kinetics, eluding so far an understanding of their origins. We present a theoretical model to explain this six-decade-old challenge by addressing the elasticity of fold surfaces of finite-sized lamella in the presence of disclination-type topological defects arising from anisotropic line tension. Entrapment of a disclination defect in a lamella results in sectors separated by walls, which are soliton solutions of a two-dimensional elliptic sine-Gordon equation. For flat square morphologies, exact results show that sectored squares are more stable than plain squares if the dimensionless anisotropic line tension parameter $\alpha ={\gamma }_{an}/\sqrt{h_4{K}_{\varphi }}$ (${\gamma }_{an}$ = anisotropic line tension, $h_4$ = fold energy parameter, $K_{\varphi }$ = elastic constant for two-dimensional orientational deformation) is above a critical value, which depends on the size of the square.

Figure 1

Base-centered orthorhombic unit cell of a crystalline polyethylene lamella. The lamellar normal is along the $z$ axis. The $c$ axis of the unit cell is tilted with respect to the $z$ axis. Shaded strips represent oriented stems formed by all-trans configurations. Folds are not shown.

Figure 2

Surface of a polyethylene lamella viewed along the $c$ axis (see Fig. 1). Stems, together with their orientation, are indicated by solid lines centered on lattice points. The shaded central region belongs to a unit cell. Two types of folds are represented by dashed lines. The two folds differ in free energy. The thick, dashed fold has the lowest free energy.

Figure 3

Illustration of an idealized, lozenge-shaped, sectored lamella bounded entirely by folds with the lowest free energy. The lamella is divided into four sectors. Folds run along the edges of the three rhombi shown.

Figure 4

Elliptic ($+1$), and hyperbolic ($-1$) disclinations in the $xy$ model. Upon traversing an anticlockwise, closed circuit, the vector field rotates in anticlockwise sense through $2\pi$ for a $+1$ disclination, and in clockwise sense through $2\pi$ for a $-1$ disclination.

Figure 5

Plot of the solution (14). Lines along rounded squares represent the fold field. Shaded regions represent the two intersecting walls, with a $+1$ disclination at the center. The outermost square is the boundary of the lamella.

Figure 6

Schematic of the surface $h(x,y)=aln[cosh(x/w)cosh(y/w\left)\right]$.

Figure 7

Regions of stability of the undeformed square (below the curve) and the sectored square (above the curve): ${\alpha }^{*}$ is the threshold value of $\alpha =w{\gamma }_{an}/K_{\varphi }$ that separates the regions of stability of the ground state, and sectored configurations. These regions are indicated by an ordinary shaded square, and a crossed, shaded square respectively. $L$ is in units of the wall width $w$, ${\gamma }_{an}$ is the anisotropic line tension, and $K_{\varphi }$ is the elastic constant for two-dimensional deformation in the nematic director.