The relative crystallinity function f(α) can be derived in the form of Avrami-Erofeev model by considering the following:[79-82]
- formation of crystal nuclei at time τ (that occupies a volume V(t, τ) at time t);
- rate of crystal nuclei growth represented by ; and
- the occurrence of randomly distributed nuclei (crystal morphology).
As per the above constitutive framework, the total volume of material Vtotal(t), that crystallizes from a given molten polymer mass, can be expressed by the following equation[35, 42, 82]
Equation (A1) is fundamentally a general convolution integral that comprises a rate law of nucleation , and a formalism for nuclei growth V(t, τ). In this regard, N denotes the number of active (growing) nuclei and V(t, τ) gives the volume at time t for a nucleus that became activated at time τ. In principle, Eq. (A1) describes a class of models with different nucleation and growth processes.
First, we shall consider . In the literature, several nucleation rate laws, such as first-order, instantaneous, linear, random (with unimolar decay), etc. are available.[44, 82] In this particular situation, we apply the first-order nucleation. This means that the rate of nucleation is linearly related to the difference between the number of germ nuclei N0 (the potential nucleus formation sites/defects) and the number of growing (active) nuclei N as follows
where knucl is the first-order nucleation rate constant. Integration of Eq. (A2) with N = 0 for t = 0 gives
Differentiation of Eq. (A3) with respect to t leads to the following exponential law of nucleation
We may note the following from Eq. (A4). When knucl is large, nucleation is virtually instantaneous, N = N0; and no further nuclei are generated during subsequent cooling of the molten polymer. On the other hand, when knucl is small, the rate of nucleation is approximately constant because the number of sites (N0−N) changes little. This is known as the linear law of nucleation
Now, we shall derive a formalism for V(t, τ). In this connection, we assume that the radius of nucleus r(t, τ) increases with a rate kgrow(t*) as follows
For n-dimensional isotropic nuclei growth, V(t, τ) can be expressed as
where Ks is the shape factor for the growing nuclei.
Substitution of Eqs. (A7) and (A4) into Eq. (A1) yields
Equation (A8) applies to any type of thermal history. Let us assume that kgrow (t*) ≠ f(t*), and consider an isothermal case; then, kgrow represents a time-invariant nucleation growth constant. Under these two conditions, Eq. (A8) becomes
The analytical solution of Eq. (A9), following successive integration by parts, can be written as
Division of both sides of Eq. (A10) by the initial volume of the molten polymer V0, yields the extended relative crystallinity αex(t) as follows
So far, the nuclei have been allowed to grow in an unlimited fashion (ignoring the constraints of overlap and site ingestion) as per the exponential law of nucleation, that is, Eq. (A4). However, the growing nuclei will eventually reach the boundaries of the crystallizing polymer; the nuclei boundaries will overlap; and they may also ingest the germ nuclei. To account for these phenomena, αex(t) is to be converted to the true relative crystallinity α(t) (fractional extent of crystallization), using the following relation proposed by Avrami for locally random distribution of germ nuclei[79-82]
Substitution of Eq. (A12) into Eq. (A11) gives
We have two time scales. One is the macroscopic time scale for the overall crystallization process, and the other is the microscopic time scale for nucleation. In practice, t >> τ, that is, all nucleation is completed before crystal growth. This represents a situation of site saturation.[20, 83] Then, the right hand side of Eq. (A13), for the limiting case of large knucl (high nucleation rate), can be approximated with the highest order term of the summation series, and using e−knuclt ≈ 0. This reduces Eq. (A13) to
where = overall crystallization rate constant.
Erofeev derived analogous material transformation expression from completely different considerations. Despite the conceptual difference, Erofeev's probabilistic approach inherently includes the same assumptions as Avrami's treatment. Therefore, Eq. (A14) is the well-known Avrami-Erofeev equation for isothermal crystallization, which yields the following relation
Differentiation of Eq. 14 gives the following rate of crystallization
The isothermal crystallization rate can also be expressed in terms of the relative crystallization function f(α(t)) as follows
Equations (A14), (A15), (A16) can be combined to give the final isothermal Avrami-Erofeev relative crystallinity function
Therefore, the isothermal Avrami-Erofeev polymer crystallization rate can be written as
Equation (A19) shows that for a given value of n, the isothermal polymer crystallization rate is a simple function of temperature (through k), and the volume fraction transformed.
Differentiating both sides of Eq. (A18) with respect to α(t) and equating the same to zero, αmax can be obtained in the form of the following expression
where αmax, a function of only n, is the value of relative crystallinity corresponding to which the isothermal relative crystallinity function f(α(t)) has a maximum.
Appendix B: Calculation of Lamellar Thickness Distribution using Constant Heating Rate DSC Experiment
PEs, including other members of polyolefins, are semicrystalline materials. Considering an orthogonal frame work, and chain-folding mechanism of crystallization, the melting point Tm and the corresponding dimensions of a crystal lamella (crystallite) can be thermodynamically related as follows through the traditional Gibbs-Thomson equation[27, 28]
where is the equilibrium melting temperature of a crystal of infinite thickness; σ1, σ2, and σ3 are the basal specific surface free energies of the crystallite, and they are associated with the energy of chain folding during the crystallization process; L1, L2, and L3 are the corresponding dimensions along the three orthogonal directions. is the heat of fusion per unit volume for the crystalline phase (perfect crystallite). Now, let us assume the following:
- The lateral and transverse dimensions of the folded-chain crystallite in bulk are much larger than the lamellar thickness. This means that L2 and L3 >> L1, which leads to L1 = Lfclamella (folded crystal lamellar thickness), and σ1 = σssfe (crystallite specific surface free energy).
- Over the temperature range under consideration, the crystallite parameters do not depend on temperature T and lamellar thickness Lfclamella.
Then Eq. (B1) reduces to the following expression[28-30]
Equation (B2) shows that Tm decreases as Lfclamella decreases; the converse is also true; and Tm → as Lfclamella → ∞. Therefore, an asymptotic relation holds between Tm and Lfclamella.
Now we shall show how to calculate the distributions of lamellar thickness and chain fold length, using a typical DSC endotherm. For this purpose, we assume that the rate of heat flow, at a given temperature, is proportional to the mass of a crystallite (having a lamellar thickness between Lfclamella and Lfclamella + dLfclamella, that has melted during dT.[28, 29] Accordingly, the melted crystallite mass fraction χi at time ti and temperature Ti is given by
Note that all the integrals in Eq. (B2) is automatically generated as a function of time and temperature by the software of a standard computer-assisted DSC instrument.