Theoretical Aspects of Polymer Crystallization

Abstract

Semicrystalline morphology of polymers is determined by the thermodynamic and kinetic factors in crystallization. In this chapter, we made a brief survey on our current understanding about thermodynamics and kinetics of polymer crystallization. On thermodynamics, we introduced the melting point, metastable states, phase diagrams, mesophase formation, as well as those structural factors governing melting points, mainly on the basis of the mean-field lattice statistical theory of crystallizable polymer solutions. On kinetics, we introduced crystal nucleation, crystal growth and crystal annealing, mainly on the basis of the classical nucleation theory and the related kinetic equations.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

4.1 Introduction

A vast amount of natural and synthetic polymers are semicrystalline . From this standpoint alone, it is necessary to understand the process of polymer crystallization, which dominates the structure formation of semicrystalline polymers and thus allows us to control their morphologies and properties. Polymer crystallization is influenced by many factors , such as chemical structures, compositions, temperatures, and thermal history. The theoretical models serve as a powerful tool for us to comprehend these factors in polymer crystallization.

Thermodynamics and kinetics are the two most fundamental theoretical aspects of polymer crystallization. Thermodynamics addresses why, or under which circumstances, polymer crystallization will begin, or in the opposite direction polymer crystals will start to melt. Kinetics addresses how fast polymer crystallization will be initiated, be developed, and be further improved. Both aspects decide crystal morphologies that eventually influence the properties of polymer materials.

This chapter intends to make a brief survey on our current theoretical understanding about the basic thermodynamics and kinetics of polymer crystallization. On thermodynamics, we will introduce the melting point, metastable states, phase diagrams, mesophase formation, as well as those factors governing melting points. On this aspect, the mean-field lattice theory on the statistical thermodynamics of crystallizable polymer solutions will be focused. On kinetics, we will introduce the basic knowledge about crystal nucleation, crystal growth, and crystal annealing. On this aspect, the classical nucleation theory and the related kinetic equations will be focused.

Due to length limitation of this chapter, many theoretical approaches on the phenomenological aspects of polymer crystallization have to be skipped. The isothermal and non-isothermal kinetic analysis of overall crystallization appears as technically important in the data treatment of DSC measurements. Some theoretical considerations on the metastable aspects of crystal morphologies and their evolution under various circumstances appear as practically important and case sensitive (see Chap. 1). In this sense, a combination of this chapter with other contributions of this book will provide reader a broad cutting-edge knowledge about our basic understanding of polymer crystallization.

4.2 Thermodynamics of Polymer Crystallization

4.2.1 Basic Concepts

Polymers are either in the fully amorphous state or in the fully crystalline state, or in the states partial between them. When they are in the amorphous state of homogeneous solutions or melt, polymer chains are fully disordered, as described by a random-coil model. The random-coil model was first proposed by Kuhn (1934) as well as by Guth and Mark (1934) to predict the entropic elasticity of polymer chains, and then was used to describe the amorphous state of polymers by Flory (1953) and others.

Polymer crystallization and melting are typically first-order phase transitions between the amorphous phase and the crystalline phase . When these two phases are in thermodynamic equilibrium, two phase transitions are thermodynamically reversible under a certain temperature. This temperature is referred to the equilibrium melting point of polymer crystallization. The free energy changes of amorphous phase and crystalline phase under various temperatures are depicted in Fig. 4.1, illustrating the definition of the equilibrium melting point T ⁰_m .

Fig. 4.1

Schematic free energy curves of amorphous phase and crystalline phase versus temperature. The temperature at which two curves intersect with each other is defined as the equilibrium melting point of polymers

In the bulk system of pure polymers, the free energy change of melting becomes zero when the system is under its equilibrium melting point, as given by

$$ \Delta {F}_{\mathrm{m}}=\Delta {Q}_{\mathrm{m}}-{T}_{\mathrm{m}}^0\Delta {S}_{\mathrm{m}}=0 $$

(4.1)

The equilibrium melting point can accordingly be calculated by

$$ {T}_{\mathrm{m}}^0=\Delta {Q}_{\mathrm{m}}/\Delta {S}_{\mathrm{m}} $$

(4.2)

In practice, to get over the nucleation barrier via thermal fluctuations , the initiation of primary crystal nucleation requires a crystallization temperature T _c something lower than T ⁰_m , and thus a supercooling for crystallization is defined as

$$ \Delta T={T}_{\mathrm{m}}^0-{T}_{\mathrm{c}} $$

(4.3)

In order to overcome the kinetic barrier under a certain supercooling , polymer crystallization commonly chooses a pathway favoring its kinetics, which will result in metastable semicrystalline states (Cheng 2008). It took a long time for people to figure out the structure features of the metastable semicrystalline polymers, with two dominant models, i.e., fringed-micelle and chain-folding models , as illustrated in Fig. 4.2. Here, the metastable state holds either a thermodynamic meaning as its local minimum in the free energy landscape, or a dynamic meaning with its negligible changes over the time window of our observations upon a very slow evolution towards more stable states.

Fig. 4.2

Illustration pictures on the models of (a) fringed-micelle (Hermann et al. 1930) and (b) adjacent chain-folding (Keller 1957)

In 1930, Hermann et al. set up the fringed-micelle model (Hermann et al. 1930; Hermann and Gerngross 1932) to describe the high elasticity of low-density polyethylene products. But later on, this model could not explain the spherulite morphologies of polymer crystals often observed under optical microscopy. In 1957, Keller set up the adjacent chain-folding model (Keller 1957) on the basis of the facts that polymer stems in single lamellar crystal grown from dilute solutions were perpendicular to the lamellar surface, and the lamellar thickness was only in the scale of one tenth of chain lengths. This model was then confirmed by the small-angle neutron scattering (SANS) experiments of single crystal grown in dilute solutions (Spells et al. 1980) as well as by the integer folding of short chains in lamellar crystals grown in the melt (Arlie et al. 1966, 1967; Ungar et al. 1985; Organ and Keller 1987). For lamellar crystals grown in the long-chain melt, Flory proposed the “Switchboard” model (Flory 1962), which was then developed into the interzonal Switchboard model as discussed by Mandelkern (1964). Since both adjacent chain-folding and Switchboard models have their own interpretations on the same SANS experiment data, there was a hot debate in 68^th Faraday Discussion Meeting. It is now well accepted that either model describes a certain aspect of structural features on lamellar crystals grown in the melt. The variable cluster model synthesizing both features of local chain-folding and global random-coil was discussed by Hoffman (1983). The loops, cilia, and tie molecules (Fig. 4.3) are restricted at the fold-end surfaces of lamellar crystals, which constitute the rigid amorphous phase near the crystalline region. The latter pretends to be the third phase besides the mobile amorphous phase and the crystalline phase, and appears as important in the thermal and mechanical properties of semicrystalline polymers (Wunderlich 2003).

Fig. 4.3

Illustration pictures on the models of (a) Switchboard (Flory 1962) and (b) variable clusters (Hoffman 1983)

As will be introduced in the following kinetic aspects of polymer crystallization, secondary intramolecular crystal nucleation favors chain-folding upon crystal growth, which dominates the lamellar feature of crystal morphologies. The metastable lamellar crystals intend to perform thickening into a more thermodynamically stable state upon annealing. Before they are able to reach the most stable state, their melting points appear as lower than the equilibrium melting point of infinitely large crystals. This is mainly because of a limited dimension in lamellar thickness. The lowered melting point of lamellar crystals T _m is described by the well-known Gibbs–Thomson equation , as given by

$$ {T}_{\mathrm{m}}={T}_{\mathrm{m}}^0-\frac{2{\sigma}_{\mathrm{e}}{T}_{\mathrm{m}}^0}{l\Delta h} $$

(4.4)

where σ _e is the free energy density of the fold-end surface, l is the lamellar thickness, and Δh is the heat of fusion.

A wide distribution of lamellar thickness in polymer crystals results in a broad range of melting temperature, shown as a wide melting peak in the heating curve of differential scanning calorimetry (DSC) . We pragmatically define the peak temperature as the experimentally observed melting point of polymer crystals, which could be far below the equilibrium melting point due to the limited lamellar thickness.

Crystallization under various temperatures yields different corresponding lamellar thicknesses, exhibiting different variable melting points. This phenomenon allows us to derive the equilibrium melting point at the infinitely large crystal according to the Gibbs–Thomson equation , provided that there is no annealing effect upon heating the crystals for the measurement of their melting points. However, most of metastable crystals do perform annealing behaviors upon heating towards melting. Hoffman and Weeks (1965) supposed that lamellar crystals will thicken into a metastable state with the thickness several times larger than the minimum thickness, and derived the equilibrium melting point as the crossover point of the extrapolated T _m versus T _c curve at T _m = T _c. Although the fixed folds of thickness increase appear as a big assumption, this method has been widely applied to derive the equilibrium melting points of various semicrystalline polymers in the literature.

Besides crystallization, liquid–liquid (L–L) demixing is another basic phase transition in polymer solutions, and both of them are functions of polymer concentrations and temperatures. The schematic L–L binodal and liquid–solid (L–S) coexistence curves in polymer solutions are separately shown in Fig. 4.4. The illustrated L–L binodal contains an upper critical solution temperature. Some solution binodals may contain a lower critical solution temperature, or even both.

Fig. 4.4

Schematic pictures of (a) L–L binodal curve and (b) L–S coexistence curve in polymer solutions

When the L–S curve intersects with the L–L curve in the overlapping temperature windows, both curves will be terminated at the intersection point, which is referred to the monotectic triple point. The typical phase diagram in polymer solutions is shown in Fig. 4.5.

Fig. 4.5

Schematic picture of typical phase diagrams combining both L–L binodal and L–S coexistence curves in polymer solutions

In between the amorphous phase and the crystalline phase, there sometimes occurs an important intermediate phase for polymers carrying anisotropic groups (called mesogen groups ) either on the chain backbone or on the chain branches, which is referred to mesophase. The mesogen groups become orientational-ordered under suitable thermodynamic conditions. This ordered state could be the well-known liquid crystal (LC) state, as a typical mesomorphic state between the amorphous state and the crystalline-ordered state for LC polymers.

The transition temperature from LC mesophase to melt is often named as the clearing point or the isotropization point T _i. If the LC mesophase is thermodynamically stable, it will occur in both heating and cooling curves when detected by DSC measurements. This mesophase is also referred to the enantiotropic mesophase. If the mesophase is kinetically favored due to large supercooling required for the initiation of crystallization, it will occur only in the cooling curve but not in the heating curve. This mesophase is then referred to the monotropic mesophase . The free energy plots for the two kinds of mesophases are demonstrated in Fig. 4.6.

Fig. 4.6

Schematic pictures of (a) enantiotropic mesophase and (b) monotropic mesophase

When L–S phase transition is accompanied by LC transition, the above phase diagram of polymer solution becomes more complicated as illustrated in Fig. 4.7.

Fig. 4.7

Schematic picture of typical phase diagram of polymer solutions combining L–L binodal (Iso), L–S coexistence (Cr), and L-liquid crystal-S coexistences (LC). The two arrows in the picture show the preparation methods of thermotropic liquid crystal and lyotropic liquid crystal, respectively (Keller 1992)

According to the phase diagram schematically shown in Fig. 4.7, two types of liquid crystal ordering with different preparation methods can be categorized. One is the lyotropic liquid crystal approached by changing the concentration in solutions. The other is the thermotropic liquid crystal approached by changing the temperature in the concentrated solutions or in the bulk phase. Their conventional preparation methods are schematically shown in Fig. 4.7. According to Onsager’s interpretation (Onsager 1949), the lyotropic liquid crystal results from an entropy-driven phase transition due to anisotropic excluded-volume interactions of mesogens. When the concentration of rod-like molecules becomes high enough, the space for anisotropic particles to move freely appears limited, resulting in an entropic loss. In this case, if parts of rodlike particles are aligned in parallel with each other in a domain of higher concentration, they release part of their space for the movement of other particles and thus increase the total entropy. This entropy-driven phase transition makes the ordered system stable. For the thermotropic liquid crystal , Maier and Saupe’s theory (1959) considers that the thermotropic liquid crystal is a result of orientation-dependent dispersion interactions between mesogens. With the decrease of temperature, spontaneous orientational ordering lowers the attractive potential energy. Combination of the above two theories gave a better interpretation to the orientational ordering for the formation of liquid crystals as well as a better calculation of phase diagrams (Jähnig 1979; Ronca and Yoon 1982, 1984; ten Bosch et al. 1983a, b; Khokhlov and Semenov 1985; Gupta and Edwards 1993; Lekkerkerker and Vroege 1993).

4.2.2 Statistical Thermodynamics of Polymer Crystallization

Statistical thermodynamic theories provide a powerful tool to bridge between the microscopic chemical structures and the macroscopic properties. Lattice models have been widely used to describe the solution systems (Prigogine 1957). Chang (1939) and Meyer (1939) reported the earliest work related with the lattice model of polymer solution. The lattice model was then successfully established by Flory (1941, 1942) and Huggins (1942) to deal with the solutions of flexible polymers by using a mean-field approximation, and to derive the well-known Flory–Huggins equation .

In the lattice model of polymer solutions, polymer chain is simply represented by a number of consecutively occupied lattice sites, each site corresponding to one chain unit. The rest single sites are assigned to solvents. This simple lattice treatment of polymer solutions allows a very convenient way to calculate thermodynamic properties of flexible and semiflexible polymer solutions from the statistical thermodynamic approach. By the mean-field assumption, the entropy part and the enthalpy part of partition function can be separately calculated.

Following Flory’s treatment of semiflexible polymer solutions (Flory 1956), the entropy part of partition function for polymer solutions is given by

$$ {Z}_{\mathrm{c}\mathrm{omb}}\approx {\left(\frac{n}{n_1}\right)}^{n_1}{\left(\frac{n}{n_2}\right)}^{n_2}{\left(\frac{q{z}_{\mathrm{c}}^{r-2}}{2{e}^{r-1}}\right)}^{n_2} $$

(4.5)

where n ₁ is the number of solvent molecules, n ₂ is the number of polymer chains, each chain composed of r monomers, and n = n ₁ + n ₂ r is the total lattice sites occupied by solvent and polymer chains. q is the coordination number of the lattice space. z _c is the conformational partition function defined as

$$ {z}_{\mathrm{c}}=1+\left(q-2\right) \exp \left(-\frac{E_{\mathrm{c}}}{kT}\right) $$

(4.6)

where E _c is the energy penalty for the q − 2 noncollinear consecutive bonds along the polymer chain with a reference to the collinear connection, k is Boltzmann constant , and T is system temperature.

The partition function related with the mixing enthalpy of solvent and monomers is given by

$$ {z}_{\mathrm{m}}= \exp \left[-\left(q-2\right)\frac{n_1}{n}\times \frac{B}{kT}\right] $$

(4.7)

where B is the energy change for a pair of solvent and monomer before and after mixing as defined by

$$ B={E}_{12}-\frac{E_{11}+{E}_{12}}{2} $$

(4.8)

As polymer crystals are commonly featured with parallel packing of polymer chains, the driving force for crystallization can be modeled as the parallel packing of bonds in the lattice model (Hu 2000). Assuming the energy penalty for nonparallel packing of two bonds is E _p, deviating from the ground state with parallel-packed bonds, the partition function related to nonparallel packing of neighboring bonds around a chain bond is given by Hu and Frenkel (2005)

$$ {z}_{\mathrm{p}}= \exp \left\{-\frac{q-2}{2}\left[1-\frac{2\left(r-1\right){n}_2}{qn}\right]\frac{E_{\mathrm{p}}}{kT}\right\} $$

(4.9)

Thus after integrating different parts, the total partition function of polymer solutions can be described as

$$ Z={\left(\frac{n}{n_1}\right)}^{n_1}{\left(\frac{n}{n_2}\right)}^{n_2}{\left(\frac{q}{2}\right)}^{n_2}{e}^{-\left(r-1\right){n}_2}{z}_{\mathrm{c}}^{\left(r-2\right){n}_2}{z}_{\mathrm{m}}^{r{n}_2}{z}_{\mathrm{p}}^{\left(r-1\right){n}_2} $$

(4.10)

The corresponding free energy density is calculated according to Boltzman’s relationship, as given by

$$ \begin{array}{ll}\frac{f\left(\phi \right)}{k_{\mathrm{B}}T}=& \left(1-\phi \right) \ln \left(1-\phi \right)+\frac{\phi }{r} \ln\;\phi \kern1em \hfill \\ {}& +\phi \left(-\frac{ \ln \left(qr/2\right)}{r}-\left(1-2/r\right) \ln\;{z}_{\mathrm{c}}+\left(1-1/r\right)+\left(q-2\right)\frac{B}{k_{\mathrm{B}}T}+\left(1-1/r\right)\frac{q-2}{2}\frac{E_p}{k_{\mathrm{B}}T}\right)\hfill \\ {}& \kern1em -{\phi}^2\left(\left(q-2\right)\frac{B}{k_{\mathrm{B}}T}+{\left(1-1/r\right)}^2\frac{q-2}{q}\frac{E_{\mathrm{p}}}{k_{\mathrm{B}}T}\right)\hfill \end{array} $$

(4.11)

Equation (4.11) can be used to calculate various thermodynamic properties of polymers, in particular, the equilibrium melting point. It can be used to calculate the mixing free energy and the binodal L–L curve, as well as the coexistence L–S curve in polymer solutions.

Following the protocols of molecular interactions set up in the lattice statistical thermodynamic theory introduced above, Monte Carlo simulations appear as a powerful tool in the study of both thermodynamics and kinetics of polymer crystallization (Hu and Frenkel 2005). Polymer motions in the simulations can be realized by a micro-relaxation model (Hu 1998). Its acceptance is judged by the Metropolis importance sampling algorithm (Metropolis et al. 1953), with the potential energy change in each step of micro-relaxation composed of three contributions, noncollinear connections of consecutive bonds along polymer chains (E _c) (Flory 1956), mixed pairs of polymer monomer and solvent (B) (Flory 1942; Huggins 1942), and nonparallel packing of neighboring bonds (E _p) (Hu 2000). The micro-relaxation model is highly efficient in relaxing local chain conformation and is a suitable approach for the study of polymer crystallization. Results from Monte Carlo simulations will be introduced along with theories of polymer crystallization in the following paragraphs.

4.2.3 Properties of Equilibrium Melting Points

4.2.3.1 Interaction Parameters

Although a precise measurement of the equilibrium melting point appears as a big challenge to the practical experiments, the statistical thermodynamic theory can predict equilibrium melting point on the basis of the mean-field assumption. Taking the fully ordered extended chains as the ground state, and considering bulk polymers with infinitely long polymer chains, r → ∞, n ₁ = 0, n = rn ₂, one can get the free energy of the amorphous state from (4.11),

$$ \frac{F}{nkT}=1- \ln\;{z}_{\mathrm{c}}+\frac{{\left(q-2\right)}^2}{2q}\times \frac{E_{\mathrm{p}}}{kT} $$

(4.12)

At the equilibrium melting point, the free energy of the amorphous state is equal to that of the ground crystalline state (zero), so inserting the expression of z _c, one can derive

$$ 1+\left(q-2\right) \exp \left(-\frac{E_{\mathrm{c}}}{k{T}_{\mathrm{m}}}\right)= \exp \left[1+\frac{{\left(q-2\right)}^2}{2q}\times \frac{E_{\mathrm{p}}}{k{T}_{\mathrm{m}}}\right] $$

(4.13)

Supposing that the first term “1” in the left-hand side of the equation is relatively small, one can omit it and derives the equilibrium melting point of bulk polymers, as given by

$$ {T}_{\mathrm{m}}\approx \frac{E_{\mathrm{c}}+\frac{{\left(q-2\right)}^2}{2q}{E}_{\mathrm{p}}}{k\; \ln \left(q-2\right)-k} $$

(4.14)

From (4.14), we can clearly see that a larger E _c (a higher rigidity of polymer chains) favors a higher equilibrium melting point. For polyolefins, larger substitute groups benefit their higher melting points because internal rotation of polymer chains becomes more difficult. For instance, the melting point of polyethylene (–(–CH₂CH₂) _n –) is 146 °C, the melting point of polypropylene (–(–CH₂CH(CH₃)) _n –) is 187 °C while for poly-3-methyl-1-butene (–(–CH₂CH(CH(CH₃)₂)) _n –), its melting point is 304 °C. Apparently, more rigid groups on the backbones benefit their higher melting point as well. For instance, the melting point of polyethylene (–(–CH₂CH₂) _n –) is 146 °C, while with increasing benzene-ring density on the polymer chains the melting point of poly-p-xylene (–(–CH₂–ϕ–CH₂–) _n –) is 375 °C and that of polyphenylene (–(–ϕ–) _n –) is as high as 530 °C. Another example, the melting point of polyethylene adipate (–(CH₂CH₂–OCO–C₆H₁₂–OCO) _n –) is 52 °C, while the melting point of polyethylene terephthalate (–(CH₂CH₂–OCO–ϕ–OCO) _n –) is 265 °C, and that of polyethylene naphthalene-2,6-dicarbonoxylate (–(CH₂CH₂–OCO–ϕ–ϕ–OCO) _n –)reaches the highest 355 °C. Melting point of polymers with extremely high rigidity may be even higher than their thermal degradation temperature where their melt phase cannot be reached. On the other hand, a larger E _p favors also a higher equilibrium melting point. Longer side groups do not help the compact packing of backbone chains. Taking the melting points of (CH₂-CHR) _n for example, if R group represents –CH₃, –CH₂CH₃, –CH₂CH₂CH₃, and –CH₂CH₂CH₂CH_3, respectively, the corresponding melting points are 187 °C, 138 °C, 130 °C, and −55 °C, respectively. The larger polarity of side groups also benefits the higher melting point of polymers. For instance, the melting point of polyethylene (–(–CH₂CH₂) _n –) is 146 °C, while the melting point of polyvinyl chloride (–(–CH₂CHCl) _n –) is 227 °C and that of polyacrylonitrile (–(–CH₂CHCN) _n –) is 265 °C. In the last case, the sequence regularity becomes a minor factor for the capability of crystallization.

The linear relationship between the melting points and E _p/E _c values in (4.14) is further confirmed by Monte Carlo simulations of polymers, and the results are shown in Fig. 4.8. The high consistency in the results between the lattice theory and Monte Carlo simulations validates the mean-field treatment in the lattice theory of polymer solutions.

Fig. 4.8

Melting temperature (T _m/E _c/k _B) of bulk polymers with variable E _p/E _c values. The Solid line is calculated from (4.13) and the circles are simulation results of 32-mer polymer solutions with a concentration of 0.9375 in a 32³ cubic lattice (Hu and Frenkel 2005)

4.2.3.2 Molecular Weights

In the equilibrium states of polymer crystals, the chain ends can be regarded as the crystalline defects in the infinitely large crystals formed by extended chains, which will apparently result in a depression of melting points with the increase of concentrations of chain ends, or in other words, with the decrease of chain lengths.

In the calculation of melting point of polymers with different molecular weights , Flory and Vrij (1963) divided the free energy change of melting into three parts. The first part is the free energy change for the melting of infinitely long polymers, the second part is the free energy change by introducing chain-end defects in the crystals, and the third part is the conformational entropy change when the infinitely long polymer chain is cut into the segments with limited chain length r. So the total free energy change of melting can be described as

$$ \Delta {F}_{\mathrm{m}}=r\Delta {F}_{\mathrm{u}}+\Delta {F}_{\mathrm{e}}-k{T}_{\mathrm{m}} \ln\;r=0 $$

(4.15)

ΔF _u is the free energy change of fusion, it can be calculated by

$$ \begin{array}{c}\Delta {F}_{\mathrm{u}}=\Delta {h}_{\mathrm{u}}-{T}_{\mathrm{m}}\Delta {s}_{\mathrm{u}}\\ {}\kern1.5em ={E}_{\mathrm{c}}+\frac{{\left(q-2\right)}^2}{2q}{E}_{\mathrm{p}}-{k}_{\mathrm{B}}{T}_{\mathrm{m}}\left[ \ln \left(q-2\right)-1\right]\end{array} $$

(4.16)

ΔF _e is the extra free energy change due to the existence of chain ends, which can be calculated from the equilibrium condition f = 0 in (4.11) by setting the chain length r = 2 in the melt phase.

$$ \Delta {F}_{\mathrm{e}}=\frac{\left(q-2\right)\left(q-1\right)}{2q}{E}_{\mathrm{p}}-{k}_{\mathrm{B}}{T}_{\mathrm{m}}\left( \ln\;q-1\right)-2\Delta {f}_{\mathrm{u}} $$

(4.17)

The term ln(r) represents a change of conformational entropy upon cutting. Inserting (4.16) and (4.17) into (4.15), the equilibrium melting point predicted by the Flory–Vrij theory can be calculated.

One can also directly calculate the equilibrium melting point from (4.11). Under equilibrium melting temperature,

$$ {\mu}^c={\mu}^s $$

(4.18)

where μ ^c and μ ^s are the chemical potentials of polymers in crystals and in solutions, respectively. We assume that μ ^c = 0 by omitting any disorder defects in the crystals. When the chemical potentials in solid and liquid are equal, we can obtain the equation

$$ \begin{array}{l}\left(1-r\right)\frac{n_2r}{n}+ \ln \frac{qn}{2{n}_2}+\left(r-2\right) \ln \left[1+\left(q-2\right) \exp \left(-\frac{E_{\mathrm{c}}}{k_{\mathrm{B}}{T}_{\mathrm{m}}}\right)\right]\\ {}\kern1em =\frac{\left(r-1\right)\left(q-2\right)}{2}\left[1-\frac{2\left(r-1\right){n}_2\left(n+{n}_1\right)}{q{n}^2}\right]\frac{E_{\mathrm{p}}}{k_{\mathrm{B}}{T}_{\mathrm{m}}}+\frac{r{n}_1^2}{n^2}\frac{\left(q-2\right)B}{k_{\mathrm{B}}{T}_{\mathrm{m}}}\end{array} $$

(4.19)

The equilibrium melting point of polymers can be calculated by taking into account all the parameters and solving (4.19).

Figure 4.9 compares the melting temperatures of bulk polymers with different chain lengths derived from the lattice theory, the Flory–Vrij equation and Monte Carlo simulations , respectively. The dimension of the temperature unit is reduced as E _c/(k _B T _m). Although they hold various assumptions, their agreements are satisfying.

Fig. 4.9

Melting temperature (T _m/E _c/k _B) of polymers with different chain length. The Solid line is calculated from (4.19) and dashed line is results of Flory–Vrij equation. The Circles are results from Monte Carlo simulations of polymer solutions with a 0.9375 concentration in a 32³ cubic lattice for short chains and a 64³ cubic lattice for long chains (Hu and Frenkel 2005)

4.2.3.3 Comonomer Contents in Random Copolymers

Real polymers are not structurally uniform along the sequences of the chain, and more or less there exist various kinds of irregularities, such as different chemical compositions, different geometrical connections, or different stereo optical isomers of monomers. Because of the spatial mismatch between irregular and regular sequences for the compact packing in crystalline order, the irregular sequences will result in a lowered melting point. If the regular sequences are referred to monomer A and the irregular sequences are referred to comonomer B, Flory (1955) considered this AB random copolymer as an ideal solution and its chemical potential deviation from the sequence-uniform homopolymer A is

$$ {\mu}_{\mathrm{A}}^{\mathrm{co}}-{\mu}_{\mathrm{A}}^0=-R{T}_{\mathrm{m}} \ln\;{X}_{\mathrm{A}} $$

(4.20)

Thus, the chemical potential change of crystallization is

$$ {\mu}_{\mathrm{A}}^{\mathrm{c}}-{\mu}_{\mathrm{A}}^{\mathrm{c}\mathrm{o}}=\Delta {H}_{\mathrm{u}}-{T}_{\mathrm{m}}\Delta {S}_{\mathrm{u}} $$

(4.21)

The subscript “u” represents the unit of mole chain monomers .

For homopolymer composed of monomer A, the chemical potential change of crystallization is

$$ {\mu}_{\mathrm{A}}^c-{\mu}_{\mathrm{A}}^0=\Delta {H}_{\mathrm{u}}^0-{T}_{\mathrm{m}}^0\Delta {S}_{\mathrm{u}}^0=0 $$

(4.22)

Considering $\Delta {H}_{\mathrm{u}}^0\approx \Delta {H}_{\mathrm{u}}$ and $\Delta {S}_{\mathrm{u}}^0\approx \Delta {S}_{\mathrm{u}}$, the melting point of copolymers can be derived as

$$ \frac{1}{T_{\mathrm{m}}}-\frac{1}{T_{\mathrm{m}}^0}=\frac{R}{\Delta {H}_{\mathrm{u}}} \ln\;{X}_{\mathrm{A}} $$

(4.23)

The similar equation for stereo-optical-isomer copolymers with comonomers homogeneously distributed in the crystallites was also given by Coleman (1958).

Flory assumed all the comonomers staying only in the amorphous phase. By considering comonomers coexisting with monomers in the crystallites, Colson and Eby (1966), and later on Sanchez and Eby (1975) gave another expression for melting point of random copolymers, as given by

$$ {T}_{\mathrm{m}}={T}_{\mathrm{m}}^0\left(1-\frac{\Delta {H}_{\mathrm{B}}}{\Delta {H}_{\mathrm{u}}}{X}_{\mathrm{B}}\right) $$

(4.24)

In (4.24), ΔH _B is the heat of fusion for each comonomer as a defect in the crystalline phase, and X _B is the mole fraction of comonomers.

4.2.4 Phase Diagrams of Polymer Solutions

From (4.11), the theoretical curves of L–S coexistence curve and L–L binodal curve can be separately calculated, provided by the absence of each counterpart.

At the equilibrium melting point, the chemical potential of polymers in the amorphous phase μ ^s is equal to that in the crystalline phase μ ^c, thus

$$ {\mu}^s-{\mu}^0={\mu}^c-{\mu}^0 $$

(4.25)

where μ ⁰ is the chemical potential of the ground state where polymer chains are totally extended and packed in parallel with each other. Since in the crystalline phase, μ ^c is almost the same as μ ⁰, we can thus derive that

$$ {\mu}^s-{\mu}^0=\frac{\partial \Delta {F}^s}{\partial {n}_2}=-kT\frac{\partial \ln\;Z}{\partial {n}_2}\approx 0 $$

(4.26)

It means

$$ \frac{\partial \ln\;Z}{\partial {n}_2}=0 $$

(4.27)

Thus, the equilibrium melting point can be calculated by solving the above equation. Changing the polymer concentrations, one can obtain the theoretical phase diagram of L–S coexistence.

The L–L binodal curve can be calculated by the chemical potential equivalence of components between the dense phase and the dilute phase after phase separation. That is,

$$ \left\{\begin{array}{c}\hfill \Delta {\mu}_{1\mathrm{a}}=\Delta {\mu}_{1\mathrm{b}}\hfill \\ {}\hfill \Delta {\mu}_{2\mathrm{a}}=\Delta {\mu}_{2\mathrm{b}}\hfill \end{array}\right. $$

(4.28)

where the subscripts “a” and “b” are used to represent the dense and dilute phases, respectively. Before reaching the final result, the mixing free energy as defined as the free energy change of polymer solutions from that of bulk amorphous polymers before mixing should be calculated, as given by

$$ \begin{array}{ll}\frac{\Delta {F}_{\mathrm{mix}}}{kT}& =\frac{F_{\mathrm{solution}}-{F}_{\mathrm{bulk}}}{kT}=-\left( \ln\;Z- \ln\;{Z}_{n_1=0}\right)\hfill \\ {}& ={n}_1 \ln\;{\phi}_1+{n}_2 \ln\;{\phi}_2+{n}_1{\phi}_2\left[\left(q-2\right)\frac{B}{kT}+\left(1-\frac{2}{q}\right){\left(1-\frac{1}{r}\right)}^2\frac{E_{\mathrm{P}}}{kT}\right]\hfill \end{array} $$

(4.29)

One can calculate the chemical potentials from (4.29) and insert them into (4.28), the equilibrium concentrations of dense and dilute phases under a certain temperature can be calculated. By changing temperature, the L–L binodal curve is thus obtained.

Monte Carlo simulations can calculate the phase diagrams of polymer solutions in a different way. In the lattice model of polymer solutions, each step of micro-relaxation in Monte Carlo simulations is determined by Metropolis sampling method with a potential energy barrier as described as

$$ \frac{\Delta E}{kT}=\frac{c{E}_{\mathrm{c}}+p{E}_{\mathrm{p}}+bB}{kT}=\left(c+p\frac{E_{\mathrm{p}}}{E_{\mathrm{c}}}+b\frac{B}{E_{\mathrm{c}}}\right)/\frac{kT}{E_{\mathrm{c}}} $$

(4.30)

where c, p, and b are the numbers of net changes in noncollinear connection, nonparallel packing and mixing pairs of polymer unit and solvent after and before each step, respectively. Figure 4.10 shows the parallel results of liquid–liquid demixing curves and liquid–solid coexistence curves in polymer solutions with different energy parameter sets, obtained from the lattice mean-field theory and Monte Carlo simulations. Theoretical calculations and Monte Carlo simulations agree well with each other, again validating the mean-field theory.

Fig. 4.10

Theoretical calculation (a) and Monte Carlo simulations (b) of liquid–liquid demixing curves (dashed lines) and liquid–solid curves (solid lines) of polymer solutions with different energy parameter combinations, denoted by T (E _p/E _c, B/E _c) (Hu et al. 2003a)

In Fig. 4.10, one can clearly see that the L–S curve is mainly determined by E _p/E _c, and the L–L curve is mainly controlled by B/E _c but is also slightly affected by E _p/E _c. By changing the values of E _p/E _c and B/E _c, the interplay between crystallization and liquid–liquid demixing can be studied in a combination of the lattice theory and parallel molecular simulations (Hu and Frenkel 2004; Ma et al. 2007, 2008), which has been introduced in the author’s book (Hu 2013).

In the practical processing of polymers, many organic small molecules such as plasticizers, anti-UV agents and releasing agents are added as diluents, which also result in a depression of melting points. As deduced from Flory–Huggins equation, the chemical potential change in polymer solution is

$$ {\mu}_2^L-{\mu}_2^0=R{T}_{\mathrm{m}}\left[ \ln\;{\phi}_2-\left(r-1\right){\phi}_1+r\chi {\phi}_1^2\right]\approx rR{T}_{\mathrm{m}}\left(-{\phi}_1+\chi {\phi}_1^2\right) $$

(4.31)

The chemical potential change upon melting is

$$ {\mu}_2^s-{\mu}_2^0\approx r\Delta {H}_{\mathrm{u}}\left(1-\frac{T_{\mathrm{m}}}{T_{\mathrm{m}}^0}\right) $$

(4.32)

ΔH _u is the melting enthalpy of polymers per mole of monomers. When the amorphous phase and the crystalline phase become in equilibrium, their chemical potentials are equal. By combining (4.31) with (4.32) and making them equal to each other, the melting point of polymers in solution is derived as

$$ \frac{1}{T_{\mathrm{m}}}-\frac{1}{T_{\mathrm{m}}^0}=\frac{R}{\Delta {H}_{\mathrm{u}}}\left({\phi}_1-\chi {\phi}_1^2\right) $$

(4.33)

This equation was fitting well with experimental results (Prasad and Mandelkern 1989). It was also verified by the results of Monte Carlo simulations (Hu et al. 2003a). Figure 4.11 shows the linear relationship according to (4.33), with the data points adopted from Fig. 4.10b.

Fig. 4.11

Rescaled data from Fig. 4.10b according to (4.33). The bulk equilibrium melting temperature (E _c/k _B T ⁰_m ) is chosen to be approximately 0.2. Lines are linear regressions of symbols at the same values of B/E _c as labeled (Hu et al. 2003a)

4.3 Kinetics of Polymer Crystallization

4.3.1 Crystal Nucleation

Polymer crystallization can be roughly divided into two sequential processes : crystal nucleation and crystal growth. When the sizes of ordered domains generated by thermal fluctuations become so large that the trend to increase the surface free energy can be overcome by the trend to decrease the body free energy, larger domains intend to be more stable (Kelton 1991). The critical sizes and the related free energy barriers depend on the crystallization temperature. Therefore, a certain supercooling is required for crystal nucleation. Considering nucleus as a sphere with a radius r, the free energy change of nucleation can be estimated as

$$ \Delta G=-\Delta g\times \frac{4}{3}\pi {r}^3+\sigma \times 4\pi {r}^2 $$

(4.34)

Δg is the melting free energy of unit volume, and σ is the specific surface free energy. Schematic plot for the free energy change with the increasing radius of nucleus is shown in Fig. 4.12,

Fig. 4.12

Schematic curve of the free energy change as the radius of nucleus formed in the amorphous phase

Δg in (4.34) is calculated as

$$ \Delta g=\Delta h-{T}_{\mathrm{c}}\Delta s\approx \Delta h-{T}_{\mathrm{c}}\frac{\Delta h}{T_{\mathrm{m}}}=\Delta h\frac{T_{\mathrm{m}}-{T}_{\mathrm{c}}}{T_{\mathrm{m}}}\propto \Delta T $$

(4.35)

Generally speaking, there are three different types of nucleation according to various dimensions, i.e., primary nucleation, secondary nucleation, and tertiary nucleation. Primary nucleation is a nucleus newly formed by thermal fluctuations, with six extra nucleus surfaces if the nucleus is considered to be cubic. Secondary nucleation is two-dimensional nucleation on the advancing surface of nucleus, with four extra surfaces produced. Secondary nucleation is easier than primary nucleation as its free energy barrier is lower. Tertiary nucleation is one-dimensional nucleation at the step edge of the spreading layer on the advancing surface of nucleus, with only two extra surfaces produced. Tertiary nucleation is so fast that it can rarely be observed. The schematic pictures of different types of nucleation are shown in Fig. 4.13.

Fig. 4.13

Schematic pictures of (a) primary nucleation, (b) secondary nucleation, and (c) tertiary nucleation

Primary nucleation is the most observable phenomenon for the initiation of polymer crystallization, which can be categorized further into homogeneous nucleation and heterogeneous nucleation. In homogeneous nucleation, polymer nuclei can be treated as a cylindrical bunch of stems due to the anisotropic molecular structure, as depicted in Fig. 4.14.

Fig. 4.14

Schematic picture of nucleus treated as cylindrical bunch of stems

Thus its free energy change during nucleation is

$$ \Delta G=-\pi {r}^2l\Delta g+2\pi rl\sigma +2\pi {r}^2{\sigma}_{\mathrm{e}} $$

(4.36)

Here, r and l are the radius and the length of the cylinder, respectively; σ is the specific free energy on the lateral surface, and σ _e is the specific free energy on the end surface.

By taking the minimum of ΔG with respect to r and l, the critical free energy barrier for nucleation is derived as

$$ \Delta {G}^{*}=\frac{8\pi {\sigma}^2{\sigma}_{\mathrm{e}}}{\Delta {g}^2}\propto \Delta {T}^{-2} $$

(4.37)

The critical sizes are separately calculated as

$$ {r}^{*}=\frac{2\sigma }{\Delta g}\propto \Delta {T}^{-1} $$

(4.38)

$$ {l}^{*}=\frac{4{\sigma}_{\mathrm{e}}}{\Delta g}\propto \Delta {T}^{-1} $$

(4.39)

So the length-to-radius ratio of the critical nucleus is

$$ \frac{l^{*}}{r^{*}}=\frac{2{\sigma}_{\mathrm{e}}}{\sigma } $$

(4.40)

For polymers, homogeneous nucleation can be realized through two typical ways. One is the so-called intramolecular nucleation (Wunderlich 1976) featured with adjacent chain-folding, which can be called as chain-folding nucleation. The other is the so-called intermolecular nucleation composed of parallel stacking among neighboring chains, which can be called as fringed-micelle nucleation. Schematic illustration of these two models can be found in Fig. 4.15.

Fig. 4.15

Schematic pictures of (a) intermolecular nucleation and (b) intramolecular nucleation

Let us first consider the intramolecular nucleation. The specific free energy on the lateral surfaces of PE crystals was estimated to be 11.8 erg/cm², and its specific free energy on the fold-end surface was about 90 erg/cm² (Hoffman and Miller 1997). By (4.40), the optimized aspect ratio of critical nucleus is 15.3. Now look at the intermolecular nucleation. The end surface has an extra free energy of about 245 erg/cm² as estimated by Zachmann (1967, 1969), due to the entropy loss of disordered chains. So the aspect ratio of critical nucleus in intermolecular nucleation is as high as 56.8, and it appears very difficult to produce such kind of fibril nucleus through thermal fluctuations for primary nucleation. Moreover, by (4.36), the higher end-surface free energy results in a higher nucleation barrier than the chain-folding model. So from both kinetics aspects of the critical nucleation barrier and of thermal fluctuations, the intramolecular nucleation is preferred in the process of primary nucleation .

Heterogeneous nucleation is primary crystal nucleation on the foreign surfaces of other materials such as catalysts, dusts, and container walls. Because less extra surface free energy is required, heterogeneous nucleation is much easier than homogeneous nucleation. Assuming that heterogeneous nucleus is a cubic nucleus, its free energy barrier is similar like homogeneous nucleation, again depending upon inverse square supercooling. When the foreign surface free energy is close to that of polymer crystals, the heterogeneous nucleation is more like a layer-by-layer growth on the foreign surface. We assume that a, b ₀, and l are its width, depth, and length, respectively, as shown in Fig. 4.16, the free energy change of nucleation is given by

Fig. 4.16

Schematic picture of heterogeneous nucleation

$$ \Delta G=-a{b}_0l\Delta g+2{b}_0l\sigma + al\Delta \sigma +2a{b}_0{\sigma}_{\mathrm{e}} $$

(4.41)

In (4.41),

$$ \Delta \sigma =\sigma +{\sigma}_{\mathrm{cs}}-{\sigma}_{\mathrm{ms}} $$

(4.42)

σ _cs is the surface free energy between crystal nucleus and substrate, and σ _ms is the surface free energy between melt and substrate. If the free energy of substrate is almost the same as crystal nucleus, Δσ ≈ 0, by minimizing ΔG of a and l separately, the critical Gibbs free energy is

$$ \Delta {G}^{*}=\frac{4{b}_0\sigma {\sigma}_{\mathrm{e}}}{\Delta g}\propto \Delta {T}^{-1} $$

(4.43)

There is also another type of primary nucleation called self-nucleation investigated first by Blundell et al. (1966). The foreign surfaces for self-nucleation are provided by crystals of the same species which survived during thermal history. Since there is no extra surface free energy change during self-nucleation, it is also called athermal nucleation . This type of nucleation is an important source of memory effects for polymer crystallization.

The nucleation rate is dominated by two factors. One is the critical free energy barrier of nucleation. Its exponential dependence was first proposed by Volmer and Weber (1926). The other is the diffusion energy barrier for molecules crossing over the liquid–solid interfaces. Its exponential dependence was first proposed by Becker and Döring (1935). The quantitative expression of the prefactor in the kinetic equation of the nucleation rate is given by Turnbull and Fisher (1949) as

$$ I={I}_0 \exp \left(-\frac{\Delta E+\Delta {G}^{*}}{kT}\right) $$

(4.44)

where ΔE is the activation barrier for short-range diffusion over the liquid–solid boundary, and I ₀ is the prefactor. The critical free energy barrier is proportional to the inverse square supercooling of primary nucleation. So when temperature is high with a high free energy barrier, the nucleation rate is small. However, when temperature is low with a high activation barrier for polymer diffusion, the nucleation rate is again small. Thus, the temperature-dependence curve of the nucleation rate is somewhat like a bell shape between the glass transition temperature and the melting temperature, as illustrated in Fig. 4.17.

Fig. 4.17

Schematic illustration on the temperature dependence (a) of the critical free energy barrier and the activation barrier for diffusion, and (b) of the bell-shape curve of the nucleation rates

At high temperatures, the kinetic studies of polymer crystallization is mostly focused on the nucleation. In this region, heterogeneous nucleation takes place and the resulted mechanical properties of the semicrystalline polymers are usually hard and brittle. While in the region of low temperatures, as a result of high density of small crystallites, the semicrystalline polymers become soft and tough.

Intramolecular nucleation is preferred in polymer crystal nucleation, both primary and secondary. The secondary intramolecular nucleation explains why chain-folding is a kinetic preference, which results in lamellar shapes during crystal growth. The typical intramolecular nucleation was investigated by Monte Carlo simulations of crystal nucleation of a single-chain system (Hu et al. 2003b). Taking the extended single chain in a crystal composed of extended parallel polymer chains as the ground state, and assuming the number of melting bonds n, the free energy change of the chain is

$$ \Delta F=\Delta fn+\sigma {\left(N-n\right)}^{2/3} $$

(4.45)

where Δfn is the bulk free energy change and σ(N − n)^2/3 is the surface free energy change. The free energy change of one bond during melting is

$$ \Delta f=\frac{q-2}{2}{E}_{\mathrm{p}}-kT\; \ln \left(q-1\right) $$

(4.46)

The first term on the right-hand side means q − 2 parallel bonds around the bond in the ground state, and the denominator “2” is the symmetric factor. The number of total conformation of a chain with n melting bonds is (q − 1)ⁿ, so the second term is the average conformational entropy change of each bond during melting.

When the system is in equilibrium, the free energy in the disordered state is equal to that in the ordered state, then

$$ \Delta {f}_{\mathrm{e}}=\sigma\;{N}^{-1/3} $$

(4.47)

And the equilibrium free energy barrier for primary nucleation is (Fig. 4.18)

Fig. 4.18

Schematic free energy of single chain with number of melting bonds under thermal equilibrium state (Hu et al. 2003b)

$$ \Delta {F}_{\mathrm{e}}=\frac{4{\sigma}^3}{27\Delta {f}_{\mathrm{e}}^2} $$

(4.48)

The free energy barriers for crystallization and melting of a single chain are separately calculated as follows,

$$ \Delta {F}_{\mathrm{c}}=\frac{4{\sigma}^3}{27\Delta {f}^2} $$

(4.49)

$$ \Delta {F}_{\mathrm{m}}=\frac{4{\sigma}^3}{27\Delta {f}^2}+\Delta fN-\sigma {N}^{2/3} $$

(4.50)

Equations (4.49) and (4.50) are confirmed by Monte Carlo simulations , as shown in Fig. 4.19. Single chains with different lengths will crystallize at the same temperature on cooling, but will melt at different temperatures on heating, with higher melting temperatures for higher chain lengths. In bulk polymers, the nucleation rates appear chain-length dependent, probably because the prefactor in the kinetic equation of nucleation could also be of chain-length dependence.

Fig. 4.19

Free energy curves with various crystalline bonds at the fixed temperature T = 2.174 E _p/k _B in the single chain systems with different chain lengths as labeled (Hu et al. 2003b)

It should be noted that intermolecular nucleation could coexist with intramolecular nucleation. Intermolecular nucleation is often observed with short chains, rigid chains, polymerizing chains, or when chains are stretched. Recently, upon stretching network polymers, the transition from intramolecular nucleation to intermolecular nucleation was observed in Monte Carlo simulations (Nie et al. 2013). By analyzing the probability of adjacent chain-folding of those newly formed crystallites with a size between 50 to 200 parallel packed bonds at each step of stretching, an obvious reduction was observed in its evolution curve under each temperature as shown in Fig. 4.20a. The corresponding critical strain was considered to be the transition point under which intramolecular nucleation is the favorite and above which intermolecular nucleation becomes the dominant. Compared the critical strain with the onset strain of crystallization during stretching under each temperature as shown in Fig. 4.20b, it is found that at low temperatures (the dimension-reduced T ≤ 4.0) intramolecular nucleation dominates the initiation of polymer crystallization with a smaller strain than the critical value, and when temperature becomes higher, intermolecular nucleation begins to dominate the initiation of crystallization.

Fig. 4.20

(a) Strain-evolution curves of chain-folding probability of small crystalline clusters containing 50–200 parallel packed bonds under different temperatures. The lines are vertically shifted by 0, 0.15, 0.25, 0.35, 0.45, and 0.55, respectively. (b) Comparison between the onset strains of crystallization and the critical strain for fringed-micelle nuclei under different temperatures (Nie et al. 2013)

Recently, some researchers proposed a preordered structure in the polymer melt before nucleation. Imai and coworkers attributed the preordered structures to spinodal decomposition during orientational fluctuations at low temperatures for cold crystallization of PET (Imai et al. 1993, 1994). The main proof came from the observation of small-angle X-ray scattering (SAXS) signal before wide-angle X-ray diffraction (WAXD) during isothermal crystallization at low temperatures (Terrill et al. 1998). Sirota and Herhold (1999) and Kraack et al. (2000) observed a mesophase of chain cluster at the early stage of nucleation of long alkane chains and there is almost no supercooling for nucleation although the mesophase disappears as soon as chain-folding occurs in the crystallization of long-enough chains (Sirota 2000). The idea of nucleation initiated by spinodal decomposition was then prevailing in many theoretical models. Olmsted and coworkers (1998) thought spinodal decomposition is a result of coupling between orientational-order fluctuations and density fluctuations at low temperatures. The spinodal decomposition will enhance crystal nucleation at a certain supercooling. In molecular dynamics simulations, Gee and coworkers (2005) observed that the crystallization behaviors of PVDF under 600 K and PE under 450 K in a time scale of nanoseconds appear as spinodal decomposition. Milner calculated the free energy change of PE crystal nucleation through a rotated mesophase, and it is lower than the surface free energy of orthogonal crystalline phase. He thought this could be the free energy barrier for crystal nucleation (Milner 2011). However, the prior occurrence of SAXS signal could be attributed to a limited instrument sensitivity or improper experiment treatment. Howard Wang thought that the signal like that of spinodal decomposition could be expected as a result of improper over-reduction in the empty correction (Wang 2006). Wang et al. (2000) attributed the phenomenon to instrument sensitivity of WAXD for the small number of crystallites in the early stage of crystallization. Indeed, improving the sensitivity of WAXD by four magnitudes, the difference between SAXS and WAXD disappeared at low temperatures as observed by Heeley et al. (2003). The remained difference at high temperatures can be associated to heterogeneous nucleation rather than homogeneous nucleation.

4.3.2 Crystal Growth

4.3.2.1 Secondary Nucleation Models

After primary crystal nucleation, crystals begin to grow. The crystal growth may be diffusion-controlled or interface-controlled. If crystal grows under a large supercooling in dilute polymer solutions, the growth rate is mainly controlled by a long-distance diffusion from the far-away bulk solution to the crystal surface. This is referred to the diffusion-controlled mechanism . The linear crystal growth rate is

$$ v=\frac{\mathrm{d}r}{\mathrm{d}t}\propto {t}^{-1/2} $$

(4.51)

The diffusion-controlled mechanism means that the crystal size is not linearly dependent on the growth time (Holland and Lindenmeyer 1962). In the commonly practical cases, crystal growth is controlled by the process at the advancing surface of the crystal. This is referred to the interface-controlled mechanism . When controlled by this mechanism, the linear crystal growth rate is independent of time. The interface-controlled mechanism can be further separated into three categories, i.e., secondary nucleation growth, screw dislocation growth, and surface roughing growth. The secondary nucleation is prevailing in the description of the kinetics of lamellar polymer crystal growth (Flory and McIntyre 1955; Burnett and Mcdevit 1957; Wunderlich and Cormier 1966; Wunderlich et al. 1967).

In experiments, for example, by small-angle laser scattering (SALS) on small crystallites, or by polarized light microscope (PLM) on large crystallites, it is found that the growth rate of lamellar crystals is independent of time. This behavior implies the surface-controlled mechanism for crystal growth. Thickening at the growth front is also observed in many experiments (Wunderlich and Mielillo 1968; Abo El Maaty and Bassett 2005; Mullin and Hobbs 2011). So the crystal growth process can be treated as two steps at the wedge-shaped growth front, secondary nucleation occurs first, followed with instant thickening until thickness becomes larger than the minimum thickness for further growth of lamellar crystals. Secondary nucleation dominates the temperature dependence of the growth rate, and thickening provides the driving force for crystal growth. As will be introduced below, Lauritzen-Hoffman model and its developments dominated the present understanding of growth kinetics of lamellar crystals. The intramolecular nucleation model recently provided a promising progress.

The linear crystal growth rate of lamella by lateral-surface advancing can be treated as the competition result between advancing rate and melting rate of the growth front. Taking intramolecular secondary nucleation as the rate-determining steps for both crystal growth and melting, they need to overcome the same nucleation energy barriers from mutually opposite directions. It is the difference of the two energy barriers ΔG that determines the linear growth rate, as derived according to (4.44) by

$$ v={v}_{\mathrm{growth}}-{v}_{\mathrm{melting}}={v}_{\mathrm{growth}}\left(1-\frac{v_{\mathrm{melting}}}{v_{\mathrm{growth}}}\right)={v}_{\mathrm{growth}}\left(1- \exp \left(-\frac{\Delta G}{k{T}_{\mathrm{c}}}\right)\right) $$

(4.52)

Assuming ΔG is very small as reflected by the excess lamellar thickness above the minimum, and replacing the exponential term by the first two terms of Maclaurin expansion $\left( \exp (x)=1+x+{x}^2/2+{x}^3/6+\cdots +{x}^i/i!+\cdots \right)$, we can get the linear growth rate as

$$ v\approx {v}_{\mathrm{growth}}\frac{\Delta G}{k{T}_{\mathrm{c}}}={v}_{\mathrm{growth}}\left(l-{l}_{\min}\right)\frac{b^2\Delta g}{k{T}_{\mathrm{c}}} $$

(4.53)

In (4.53), the term l − l _min determines the net free energy for crystal growth, b is the average distance between stems inside the crystal, and Δg is the free energy change for melting of unit volume. Thus, we can treat v _growth in (4.53) as the free energy barrier for crystal growth and the rest part as the driving force for crystal growth (Ren et al. 2010). Under low temperatures, l > l _min and the crystal will grow, while under high temperatures, l < l _min and the crystal will melt. So temperature variation can lead to a continuous switching between growth and melting at the lateral growth front of lamellar crystals (Ren et al. 2010).

Lauritzen–Hoffman (LH) theory is still the most widely used theoretical model in the explanation of the growth kinetics of lamellar crystals. Figure 4.21 schematically shows the basic assumptions of folded stems at the growth front in the LH theory, without considering the thickening at the lateral growth front.

Fig. 4.21

Schematic picture of folded stems at the growth front considered in the LH theory

The LH theory holds four basic assumptions as listed below.

• 1. The growth front of polymer crystal is smooth. Secondary nucleation begins with a first stem deposited at the growth front, and follows with lateral spreading until reaching the lateral edges of the front substrate.

• 2. The chain-folded length l is constant during crystal growth. The width, thickness and number of stems are separately a ₀, b ₀ and v.

• 3. The number of growth fronts holding v stems N _v is in a steady-state distribution.

• 4. Each stem should go through an activation state before entering the crystal lattice where the fraction in each stem successfully entering the crystal lattice is ϕ.

If the number of grown stems is v, the change of Gibbs free energy is

$$ \Delta G=2{b}_0l\sigma +2\left(\nu -1\right){a}_0{b}_0{\sigma}_{\mathrm{e}}-\nu {a}_0{b}_0l\Delta g $$

(4.54)

When v = 1, the growth rate for the first stem is

$$ {A}_0=\beta\;\exp \left(-\frac{2{b}_0l\sigma -\phi {a}_0{b}_0l\Delta g}{kT}\right) $$

(4.55)

The melting rate of the first stem is

$$ {B}_0=\beta\;\exp \left(-\frac{\left(1-\phi \right){a}_0{b}_0l\Delta g}{kT}\right) $$

(4.56)

If v > 1, the growth rate of the rest stems is

$$ A=\beta\;\exp \left(-\frac{2{a}_0{b}_0{\sigma}_{\mathrm{e}}-\phi {a}_0{b}_0l\Delta g}{kT}\right) $$

(4.57)

The melting rate of the rest stems is

$$ B={B}_0 $$

(4.58)

where β is a kinetic prefactor defined as

$$ \beta =\frac{kT}{h} \exp \left(-\frac{\Delta E}{kT}\right) $$

(4.59)

By the assumption of a stable distribution of the growth fronts in the steady state, the growth flux of fronts is

$$ S={N}_0\beta\;\exp \left[-\left(2{b}_0\sigma -\phi {a}_0{b}_0\Delta {g}_{\mathrm{f}}\right)\frac{l}{kT}\right]\left[1- \exp \left(\frac{2{a}_0{b}_0{\sigma}_{\mathrm{e}}-{a}_0{b}_0l\Delta {g}_{\mathrm{f}}}{kT}\right)\right] $$

(4.60)

Different growth crystals contain different folded lengths. S(l) is the crystal growth flux with folded length l. Thus, the average length of all the crystals is calculated as

$$ \left\langle l\right\rangle =\frac{{\displaystyle {\int}_{l_{\min}}^{\infty }lS(l)\mathrm{d}l}}{{\displaystyle {\int}_{l_{\min}}^{\infty }S(l)\mathrm{d}l}} $$

(4.61)

The result of (4.61) is

$$ \left\langle l\right\rangle =\frac{2{\sigma}_{\mathrm{e}}}{\Delta g}+\frac{kT}{2{b}_0\sigma}\times \frac{2+\left(1-2\phi \right){a}_0\Delta g/\left(2\sigma \right)}{\left[1-\phi {a}_0\Delta g/\left(2\sigma \right)\right]\left[1+\left(1-\phi \right){a}_0\Delta g/\left(2\sigma \right)\right]} $$

(4.62)

where 2σ _e/Δg is the minimum length for steady growth. Equation (4.62) is used to predict the average thickness of lamellar crystals grown at different temperatures.

Besides an explanation of chain-folding lengths, the phenomenon of regime transitions can also be explained by the LH theory. If crystal growth is controlled by secondary nucleation, the temperature dependence of crystal growth rates can be dominated by

$$ G={G}_0 \exp \left(-\frac{U}{T-{T}_0}\right) \exp \left(-\frac{K_g}{T\Delta T}\right) $$

(4.63)

The first exponential term is attributed to short-range diffusion across the interface and the second term is attributed to secondary nucleation with the nucleation barrier proportional to the inverse supercooling. When temperature is high enough, crystal growth rate becomes mainly controlled by secondary nucleation. In this case, the curve of lgG + U/(T − T ₀) versus −1/T/ΔT can be divided into three linear regimes with the lowering of temperature, and the ratios of K _g among three regimes are

$$ {K}_g\left(\mathrm{I}\right):{K}_g\left(\mathrm{I}\mathrm{I}\right):{K}_g\left(\mathrm{I}\mathrm{I}\mathrm{I}\right)=2:1:2 $$

(4.64)

The above ratio is the so-called regime-transition phenomenon , which can be explained on the basis of secondary nucleation assumed in the LH theory. When crystallization takes place at a very high temperature, the rate of secondary nucleation i becomes the rate-determining step, and the followed surface spreading rate g is very large, so i < g. The secondary nucleation is supposed under the determination of the free energy barrier for depositing the first stem (Lauritzen and Hoffman 1960; Hoffman and Lauritzen 1961; Hoffman et al. 1976). Once the nucleus becomes stable, it will spread to the two lateral sides very fast with a layer thickness of b. The growth front with the width L is smooth until the next nucleus shows up. This temperature range is referred to Regime I and the crystal growth rate in this regime is

$$ {G}_{\mathrm{I}}=ibL $$

(4.65)

Molecular simulations have reproduced regime-transition phenomena (Hu and Cai 2008). However, the growth front of Regime I is rather rough, favoring an alternative interpretation based on the intramolecular secondary nucleation model (Hu and Cai 2008).

When temperature is lower than Regime I, the rate of secondary nucleation becomes higher and i ~ g. Several nuclei will grow together and the growth front will no longer be smooth. This temperature range is referred to Regime II. The advancing rate is proportional to the square root of secondary nucleation rate i, as given by

$$ {G}_{\mathrm{II}}=b{(2gi)}^{1/2} $$

(4.66)

If the temperature is further lower than Regime II, secondary nucleation rate becomes much larger than the spreading rate, i > g. In this case, several crystal layers will grow at the same time, each layer with several nuclei and the distance between two nuclei is L′. This temperature range is referred to Regime III and the crystal growth rate is

$$ {G}_{\mathrm{III}}=ib{L}^{\prime } $$

(4.67)

In the LH theory , chain-folded length was assumed to be constant and thickening was omitted during crystal growth. Also the free energy barrier for secondary nucleation was assumed to be determined by the first stem, and no more stem was considered. These assumptions are not so reasonable and many other models (Armistead and Goldbeck-Wood 1992) have been proposed in order to reach a better understanding about the growth kinetics of lamellar crystals.

Wunderlich and Arakawa observed the layer structure of PE crystal under atmospheric pressures higher than 3 kbar and the crystallinity was almost 100 % (Prime and Wunderlich 1969). In the crystals, polymer chains were perpendicular to the lamella and the largest thickness was even larger than the molecular length, which indicated the existence of extended chains in the crystals (Olley and Bassett 1977). By further observation of the lamella growth front, Wunderlich found the wedge-shaped growth front and proposed a thickening-growth mechanism under high pressures (Wunderlich 1976). In his explanation, molecular nucleation or secondary nucleation firstly took place at the growth front, and later-on developed into extended-chain crystals by fast thickening. By observing the growth process of PE folded-chain crystal (FCC) and extended-chain crystal (ECC) , Hikosaka developed the growth mechanism with chain-sliding diffusion for thickening, on the basis of the LH theory (Hikosaka 1987, 1990). In his equation of nucleation rate, the free energy barrier for short-range diffusion across the interface was also considered besides the free energy barrier of critical nucleus. The growth appears as two-dimensional, which holds both lateral and longitudinal growth of the chain stems. The two-dimensional nucleation growth mechanism can be used to explain the dependence of lamella thickness on supercooling near the triple point of high pressures (Hikosaka et al. 1995).

Wunderlich and Mehta put forward the concept of molecular nucleation (Wunderlich and Mehta 1974; Mehta and Wunderlich 1975; Wunderlich 1979; Cheng and Wunderlich 1986) in order to explain the molecular weight effect in crystallization of PE and some other polymers. At very high temperatures, the fraction of high molecular weights will crystallize first. It was proposed that each molecule entered the crystal with an additional nucleation barrier. Chain-folded secondary nucleus formed by molecules with enough lengths can only be stabilized over the critical size, while short-chain nucleus will be melted again, as illustrated in Fig. 4.22. The concept of molecular nucleation can be regarded as a patch on the LH model.

Fig. 4.22

Schematic picture of molecular nucleation (Wunderlich 2005)

The free energy change of molecular nucleation at the growth front is

$$ \Delta G=-abl\Delta g+2bl\sigma +2\left(n-1\right) ab{\sigma}_{\mathrm{e}}+2 ab{\sigma}_{\mathrm{e}}^{\prime } $$

(4.68)

σ _e is the surface free energy with chain cilia, thus the free energy change of critical nucleus is

$$ \Delta {G}^{*}=\frac{4b\sigma {\sigma}_{\mathrm{e}}}{\Delta g}+2 ab{\sigma}_{\mathrm{e}}^{\prime } $$

(4.69)

This critical free energy barrier is higher than normal secondary nucleation and thus the molecular nucleation can be the rate-determining step of crystal growth.

The intramolecular nucleation was then developed by supposing that all the secondary nucleation is mainly controlled by intramolecular nucleation (Hu et al. 2003b; Hu 2007) and the basic crystal lamella is resulted due to the preference of chain-folding in this unique style of secondary nucleation.

Assuming secondary nucleation of a single chain at the two-dimensional locally smooth growth front, the free energy change based on the classical nucleation theory is

$$ \Delta F=\Delta fn+\sigma {\left(N-n\right)}^{1/2} $$

(4.70)

The free energy barrier for crystal nucleation is

$$ \Delta {F}_{\mathrm{c}}=\frac{\sigma^2}{4\Delta f} $$

(4.71)

And the equilibrium free energy barrier is

$$ \Delta {F}_{\mathrm{e}}=\frac{\sigma^2}{4\Delta {f}_{\mathrm{e}}} $$

(4.72)

The free energy barrier for critical intramolecular nucleation under a certain temperature is independent of chain length, but not for the opposite direction, i.e., melting. Therefore, a critical molecular length exists for the equilibrium intramolecular melting at the growth front under each temperature. For a polymer sample with a polydispersity of molecular weights, only the fractions of chain lengths larger than the critical length can be stable and thus enter the crystal during secondary nucleation. This is the reason why molecular segregation occurs upon lamellar crystal growth. Molecular segregation can be observed only in polymer crystallization under very high temperatures.

Under a low temperature, the critical chain length for secondary nucleation may be much smaller than the chain length, and several events of intramolecular nucleation could happen along the same chains. If they occur in the same lamellar crystal, loops are formed; and if in different lamellar crystals, tie molecules are formed. The intramolecular nucleation model allows a statistical treatment on the semicrystalline texture.

The spreading right after the event of intramolecular nucleation may be stopped by the entanglements of long chains, collisions of nuclei at the same growth fronts, or the limited width at the growth front. Monte Carlo simulations (Hu et al. 2003c) have demonstrated that a single chain enters into the crystal growth front via several events of surface nucleation along the chain, as a result of limited growth-front sizes.

Besides molecular segregation (Hu 2005), many other phenomena unique for polymer crystal growth also favor the intramolecular nucleation model, such as co-crystallization of long and short chains (Cai et al. 2008; Jiang et al. 2015) as well as the interpretation of regime transitions (Hu and Cai 2008).

4.3.2.2 Other Non-nucleation Models

There are some models based on non-nucleation mechanisms for lamellar crystal growth. The (200) growth front of PE single crystals will become curvature when crystallization temperatures are very high, which could not be explained by the LH theory based on secondary nucleation growth on smooth surface. Also a pair of concave (110) surfaces were observed in the twin single crystal, which means the free energy barrier from side surfaces may not be the main problem for the advancing of the growth front (Sadler et al. 1986). Based on these observations, Sadler and Gilmer (1988) (SG) proposed the row model of continuous growth along the direction perpendicular to the growth front, as illustrated in Fig. 4.23.

Fig. 4.23

Schematic pictures of the row model proposed by Sadler and Gilmer (1988)

In the SG model, crystal unit will be randomly added in or be removed from the growth front with a free energy change. On the one hand, the longer the growth stem, the larger the driving force for advancing the growth front. The driving force is proportional to the difference between the stem length and the critical length for a stable stem . On the other hand, the thicker the growth front, the longer time needed to reach the critical stem length because chain extending is pinned by the metastable chain-folding or loops. The possibility of stem lengths among all the conformations is only

$$ P={e}^{- kl} $$

(4.73)

Thus, the total growth rate is

$$ G\propto {e}^{- kl}\left(l-{l}_{\min}\right)\Delta g/(kT) $$

(4.74)

The stem lengths exhibit two opposite trends of contributions to the total growth rate, and the maximum of growth rate will be realized at a certain stem length, which explains the growth thickness of the lamellar crystals. The thickness can be calculated as

$$ \left\langle l\right\rangle ={l}_{\min }+\frac{1}{k} $$

(4.75)

Like in the LH theory, the free energy barrier for crystal growth in the SG model is related directly with the lamellar thickness without considering further thickening after crystal growth. Also, the proposed thermal roughening may still be flat at the crystal edge, without the necessity of curvature at the growth surface.

In both LH and SG models, the growth front was supposed to directly reach the critical thickness in the growth process, and the thickening after growth was neglected, which appears not so reasonable. Keller and coworkers (Keller 1992; Keller et al. 1994) proposed a wedge-shaped growth-front model considering an obvious thickening in the crystal growth process of PE, as illustrated in Fig. 4.24.

Fig. 4.24

Schematic picture of wedge-shaped growth front of PE lamella

In the wedge-shaped growth model, the melt may first grow into a mesophase (hexagonal phase) in the thinnest region of the growth front. The mesophase is stable because of large specific surface energy of small crystallites, which is referred as the finite-size effect. Then, the thin lamella thickens into the stable orthogonal phase, which decides the lamellar thickness. There is a triple point Q, as demonstrated in Fig. 4.25. If temperature is above the triple point temperature, the melt will grow into the orthogonal phase directly. If temperature is below the triple point, there will be mesophase.

Fig. 4.25

Schematic picture showing temperature against reciprocal thickness of polyethylene crystals

This model was later on expanded by Strobl (2000, 2005, 2006, 2009) to other polymers, in order to explain the experimental observations of his group, as illustrated in Fig. 4.26.

Fig. 4.26

Schematic picture of mesophase at the growth front in lamella (Strobl 2009)

There is a linear relationship between the crystallization temperatures and the inverse lamella thicknesses, which is quite in accordance with Gibbs–Thomson equation. There is also a linear relationship between the melting temperatures and the inverse lamella thicknesses. Crossover of these two linear curves is considered to be the triple point of mesophase transition. Recently, the crossover was reproduced in the molecular simulations of lattice polymers, and the interpretation was updated to an uplimit of instant thickening at the lateral growth front of lamellar crystals (Jiang et al. 2016).

Allegra (Allegra 1977, 1980; Allegra and Meille 1999, 2005) proposed statistical thermodynamic theory for the mesophase of small crystallites or crystal cluster in the metastable disordered phase before crystallization. The cluster will first grow into a stable size and then joins into the crystal, as illustrated in Fig. 4.27. The thickness of lamella is decided by the cluster size. Zhang and Muthukumar (2007) performed simulations of clusters to form single crystals grown in dilute solutions, consistent with the experimental observations.

Fig. 4.27

Schematic picture of equilibrium-sized cluster growth model (Allegra and Meille 2005)

Muthukumar and coworkers (Welch and Muthukumar 2001; Muthukumar 2005) gave a thermodynamic explanation to the lamella thickness. He thought that the finite lamella thickness was a result of the largest thermodynamic stability of small crystallites. The crystalline chain will find the folded length related to the minimum free energy of the whole crystallite although there is a free energy barrier for the thickening of integer folding. The mechanism for crystal growth to select a limited lamellar thickness is recently addressed by the combination of secondary nucleation and instant thickening at the lateral growth front of lamellar crystals (Jiang et al. 2016).

4.3.3 Crystal Annealing

Annealing is a procedure to keep the temperature of a crystal body near its melting point so as to relax its inner stress and to remove defects. When annealing is used for polymer materials, it can make metastable polymer crystals more perfect and more stable. If the annealing temperature is low, it just makes the crystal more perfect by removing defects from crystal lattice. As a result, crystallinity is increased. If the annealing temperature is high, crystal thickening will happen and results in more stable crystals. If the annealing temperature is very high, even higher than the melting point of metastable crystals, crystals may melt and recrystallize into a more stable state unless the temperature is close to the equilibrium melting point of infinitely large crystals.

There are two mechanisms to explain lamellar thickening, one is solid-chain sliding-diffusion mechanism , and another is melting-recrystallizaiton mechanism . Peterlin (1963) proposed an activation energy barrier for sliding diffusion of folded chains in monolayer crystals, trying to explain why the folded length of polymer chains increases linearly with the logarithm of time. The sliding-diffusion mechanism was then developed into a more general theory by Sanchez and his collaborators (1973, 1974). Dreyfus and Keller (1970) proposed the fold-dislocation thickening model, in which the lamellar thickness can be doubly increased, as schematically demonstrated in Fig. 4.28. Another is the melting-recrystallization mechanism, which was firstly reviewed by Fischer (1969) and then introduced by Wunderlich (1976) in his famous book. This mechanism was confirmed by several experimental phenomena, such as decrease-then-increase of crystallinity in the annealing process (Matsuoka 1962).

Fig. 4.28

Schematic picture of the fold-dislocation thickening model (Dreyfus and Keller 1970). (a) → (e) shows the thickening process of folded chain in a lamellar crystal. The space generated in this process should be filled by other stems (b′) or be discharged by merging of stems (b″)

The phenomenon that lamella thickness increases with the logarithm of time has been observed in many experiments (Fischer and Schmidt 1962; Wunderlich and Mielillo 1968). This continuous thickening of mobile high-molecular-weight polymer crystals was simulated by Monte Carlo simulations (Wang et al. 2012). Figure 4.29a demonstrates the wedge-shaped profile of the growth front resulted from the continuous thickening by chain-sliding diffusion at the growth front. Figure 4.29b provides three different growth-front profiles at different temperatures, which can be fitted into a logarithmic function of distances to the growth front. This function implies a logarithmic time dependence of crystal thickness. The prefactor in the function indicates an increasing thickening rate with increasing temperatures. This continuous thickening is obviously controlled by the chain-sliding diffusion in the crystals. The similar thickness profiles at the edges of lamellar polymer crystals have been observed in experiments (Reiter 2014).

Fig. 4.29

(a) Snapshot of the wedge-shaped lamellar crystal of high-molecular-weight polymers grown for 35000MCc under T = 4.6 E _c/k _b in the 64³ cubic lattice and polymer occupation density was 0.9375. Parameters were set as E _p/E _c = 1 for flexible chains and E _f/E _c = 0.02 to allow chain-sliding diffusion. The template was placed at the left end and only crystalline bonds were shown in yellow color. (b) Lamellar thickness as a function of distance to the growth front at three different temperatures T = 4.6, 4.8, and 5.0, respectively. The equations shown in the picture are the calculated logarithmic functions of distance D to the growth front (Wang et al. 2012)

The logarithmic-time dependence of crystal thickness can be deduced easily. By assuming a frictional barrier (ΔE _s) for chain-sliding diffusion proportional to the lamella thickness (l), thus the thickening rate of monolayer lamellar crystal under a certain temperature is

$$ \frac{\mathrm{d}l}{\mathrm{d}t}\propto b{e}^{- al/{k}_{\mathrm{b}}T} $$

(4.76)

Equation (4.76) can be solved by

$$ l=c\; \ln\;t+d $$

(4.77)

In the above equations, a, b, c, and d are the coefficients.

4.4 Summary

We made a brief introduction about our current theoretical models of thermodynamics and kinetics of polymer crystallization. We first introduced basic thermodynamic concepts, including the melting point, the phase diagram, the metastable state, and the mesophase. The mean-field statistical thermodynamics based on a classic lattice model of polymer solutions, and its predictions on the melting points and phase diagrams were emphasized. Those molecular factors governing the melting points were also introduced. We then introduced crystal nucleation, crystal growth, and crystal annealing as the three basic stages of polymer crystallization. The classical nucleation theory as well as some recent ideas about primary nucleation were emphasized. On crystal growth, we introduced the secondary nucleation model, in particular, the well-known Lauritzen–Hoffman theory. The recently developed intramolecular nucleation model can be regarded as an updated version of secondary nucleation models. Some other models based on non-nucleation ideas were also introduced in a balanced way. The kinetics of crystal thickening, which is usually dominating crystal annealing, was also introduced under three typical circumstances.

Due to the length limitation, we have to skip many other theoretical aspects of polymer crystallization. We hope that our selected content above has already been strong enough to demonstrate the power of theoretical approaches, as a complemental to experimental approaches to gain a better understanding of the complicated polymer crystal morphologies.