Colloquium: Laws controlling crystallization and melting in bulk polymers

After the fundamental structure of semicrystalline polymers—platelike crystallites with thicknesses in the nanometer range, embedded in a liquid matrix—was discovered in the late 1950s, attention turned to the mechanism of formation. After intense controversial discussions, an approach put forward by Hoffman and Lauritzen prevailed and was broadly accepted. The picture envisaged by the treatment—platelike crystallites with atomically smooth side faces and a surface occupied by chain folds, growing sideways layer by layer with a secondary nucleation as the rate-determining step—was easy to grasp and yielded simple relationships. The main control parameter is the supercooling below the equilibrium melting point of a macroscopic crystal $T_f^{\infty }$, which determines both the thickness of the crystallites and their lateral growth rate. The impression that the mechanism of polymer crystallization was understood and the issue essentially settled, however, was wrong. Experiments carried out during the last decade on various polymer systems provided new insights which are now completely changing our understanding of such systems. They revealed a number of laws that control polymer crystallization and melting in bulk, showing in particular that the crystal thickness is inversely proportional to the distance to a temperature $T_c^{\infty }$ which is located above the equilibrium melting point, and that crystal growth stops at a temperature $T_{\mathrm{zg}}$ which is below $T_f^{\infty }$. Observations indicate that the pathway followed in the growth of polymer crystallites includes an intermediate metastable phase. In a proposed model a thin layer with mesomorphic inner structure forms between the lateral crystal face and the melt. The first step in the growth process is attachment of the coiled chain sequences of the melt onto the mesomorphic layer, which subsequently is transformed into the crystalline state. The transitions between melt, mesomorphic layers, and lamellar crystallites can be described with the aid of a temperature-thickness phase diagram. $T_c^{\infty }$ and $T_{\mathrm{zg}}$ are identified with the temperatures of the (hidden) transitions between the mesomorphic and the crystalline phase, and between the liquid and the mesomorphic phase, respectively. Comparing predictions of the model theory with experimental results from small-angle x-ray scattering, optical microscopy, and calorimetry yields in addition to the three equilibrium transition temperatures latent heats of transition and surface free energies.

Figure 1

Electron micrograph of a carbon film replica on a surface of PE (width of the depicted region is $5\mu \mathrm{m}$). One observes stacks of platelike crystallites with curved edges and a varying orientation on the sample surface. From 3.

Figure 2

Ultrathin slice of a PE sample stained with $\mathrm{Os}{\mathrm{O}}_4$. The bright lines are crystalline lamellae of PE which are oriented edge on, i.e., with the plate surface perpendicular to the surface of the slice. Crystallites are embedded in a dark fluid matrix. From 13.

Figure 3

Crystal structure of short-chain molecules like the $n$-alkanes. Schematic drawing with two layers. The layer thickness corresponds to the length of the molecules; the distance between neighbors is about $0.5\mathrm{nm}$.

Figure 4

Part of a lamellar crystallite in a semicrystalline polymer. Parallel straight chain sequences with a length of the order $10\mathrm{nm}$ set up the crystalline structure in the interior. The two surfaces, commonly called “fold surfaces,” are occupied by sharp folds, loops, entanglements, and other noncrystallizable chain parts.

Figure 5

Growing spherulites observed during the crystallization of PLLA in a polarizing optical microscope (width of depicted region is $10\mu \mathrm{m}$).

Figure 6

Spherulite of iPS in an electron micrograph. The orientation of the lamellar crystallites varies. Those in the central parts of the lower half are standing up; those in the lower right corner are lying flat (left). Schematic drawing showing the inner structure of a spherulite resulting from branching and splaying (right). From 29.

Figure 7

Temperature dependence of the radial growth rate of spherulites of $\mathrm{P}ϵ\mathrm{CL}$. From 1.

Figure 8

Growth of a polymer crystallite as described by the Hoffman-Lauritzen model. The platelike crystallite (fold surface in dark color; the thickness here is denoted $L_c^{*}$ and the width $L$ is assumed constant) extends in one lateral direction only with a growth rate $G$. The rate-determining step is the formation of a secondary nucleus on the smooth growth face made up of refolded, repeatedly stretched chain sequences (folds connecting adjacent stems of stretched chain parts are indicated in the drawing). The attachment of further stems subsequent to the nucleation step (with rate $g$) is treated as a rapid process $(g⪢G)$. From 10.

Figure 9

sPPcO15: Results of temperature-dependent SAXS experiments. Crystallization line describing the relationship between the crystallization temperature $T$ and the inverse crystal thickness $d_c^{-1}$ (open symbols) and Gibbs-Thomson melting line giving the melting points $T$ as a function of $d_c^{-1}$ (filled symbols). The vertical direction of the connecting lines indicates that crystals have a constant thickness up to the melting point. From 6.

Figure 10

$Pϵ\mathrm{CL}$: Crystallization line and melting line. For crystallization temperatures below $40°\mathrm{C}$ crystals increase in thickness before the final melting. From 7.

Figure 11

sPPcO20: Relationship between crystallization temperature and crystal thickness in the range around the point of intersection between the melting and crystallization lines. Crystallization at the three highest temperatures (square symbols) was carried out by applying “self-seeding.” From 26.

Figure 12

sPP and $\mathrm{sPPcO}x$: Unique crystallization line (open symbols) and series of melting lines (filled symbols). Extrapolation of the melting lines to $d_c^{-1}=0$ yields the corresponding equilibrium melting points. They decrease with increasing co-unit content. From 6.

Figure 13

(Color online) Three different samples of sPP, crystallized at various temperatures and heated: Inverse crystal thicknesses at the beginning (open squares), at melting points (filled squares), and at the end point of recrystallization processes (stars). All crystallization and recrystallization lines (dots) are identical, the melting lines (dashes) are shifted against each other. From 8.

Figure 14

(Color online) General scheme treating crystallization, recrystallization, and melting, exemplified with data of sPP. Sample-invariant crystallization and recrystallization lines; sample-dependent melting line (here for sPP-Mitsui). Pathways followed during heating processes subsequent to an isothermal crystallization at low temperatures (recrystallization before melting; fixed melting point $X_s$ at the intersection of recrystallization line and melting line) and high temperatures (melting without prior recrystallization).

Figure 15

(Color online) Sample of iPP: AFM tapping mode image of lamellar crystallites which stand up, i.e., are oriented with the fold surfaces perpendicular to the image plane. The lines representing the edges of the crystallites are not continuous as in the electron micrograph of Fig. 2 but broken up in small blocks. The image demonstrates that the lamellar crystallites have a granular substructure (scan over $1\mu \mathrm{m}$ in both directions). From 16.

Figure 16

(Color online) Sample of sPP: AFM tapping mode image of the edges of up standing crystals showing a granular substructure (scan over $1.25\mu \mathrm{m}$ in both directions). From 11.

Figure 17

(Color online) Different samples of sPP (sPP, $\mathrm{sPPcO}x$, and sPP-Fina) crystallized at various temperatures $T$: Crystallization line $d_c^{-1}$ vs $T$ determined by SAXS (open symbols) and inverse lateral coherence lengths $D_{200}^{-1}$ derived from the linewidth of the 200 reflection (filled symbols). From 9.

Figure 18

$\mathrm{P}ϵ\mathrm{CL}$: Temperature dependence of the radial growth rate. Plot based on Eq. (7) giving $T_{\mathrm{zg}}=77°\mathrm{C}$. From 1.

Figure 19

Growth rates of PE represented according to Eq. (7). From 2.

Figure 20

Thermodynamic conditions assumed for crystallizing polymers: Temperature dependencies of the bulk chemical potentials of a mesomorphic (label $m$) and the crystalline phase $\left(c\right)$. The potentials are referred to the chemical potential of the amorphous melt $\left(a\right)$. From 27.

Figure 21

Multistage model of polymer crystal growth. Rather than directly attached to the crystal surface, chain segments of the melt are first incorporated in a thin layer with mesomorphic structure in front of the crystallite. The mesomorphic layer thickens spontaneously. When a critical thickness is reached, a crystal block forms by a first-order transition. In the last step the excess energy of the fold surface is reduced. From 28.

Figure 22

(Color online) $T∕n^{-1}$ phase diagram for polymer layers in a melt (label $a$) dealing with three phases: mesomorphic $\left(m\right)$, native crystalline $\left(c_n\right)$, and stabilized crystalline $\left(c_s\right)$. Lines of size-dependent phase transitions: $T_{m{c}_n}$ between mesomorphic and native crystalline layers, $T_{a{c}_n}$, $T_{m{c}_s}$, $T_{a{c}_s}$, and $T_{am}$ with corresponding meanings. Two routes for an isothermal crystallization followed by heating: route A for low crystallization temperatures and route B for high crystallization temperatures. Triple points $X_n$ $\left(X_s\right)$ with coinciding Gibbs free energies of the melt, a mesomorphic layer, and a native (stabilized) crystalline layer with the same thickness. From 27.

Figure 23

(Color online) $\mathrm{P}ϵ\mathrm{CL}$: Crystallization line (continuous), recrystallization line (dots), and melting line (dashes) determined by SAXS, zero-growth-rate temperature $T_{am}^{\infty }$ (from Fig. 18), and $a\Rightarrow m$ transition line passing through $T_{am}^{\infty }$ and $X_s$ (dash-dotted line). From 28.