T he crystal size (lamellar thickness) distribution and the average crystal size have signifi cant effects on both the processing and application properties of semi-crystalline polymers. These crystalline properties are strongly dependent on molecular structure and processing conditions. Thus, it is useful to understand and determine the relationships that govern crystalline sizes and size distributions. Furthermore, it would be both desirable and useful to solve the inverse problem of determination of the melting characteristics of a polymer with a specifi ed crystal size distribution.Because it is the simplest and fastest method, differential scanning calorimetry (DSC) has been widely used to determine the crystal size distribution. Although the melting process is a non-equilibrium process, it may be treated as a sequence of pseudo-equilibrium states, by dividing the process into a series of small steps when the scanning rate is not high. Generally, there are two methods to analyze the DSC traces. One method is to analyze DSC endothermic results directly, in conjunction with a certain relationship, such as the Gibbs-Thomson equation (Hoffman et al., 1976). The normalized heat fl ow at a given temperature in the endothermic curve is assumed to be proportional to the weight fraction of crystalline lamellae that melt at that temperature. The second method employs a differential approach, where the mass fraction of the crystalline phase is calculated by the normalized heat of fusion (Alberola et al., 1990).In this paper, we examine the relationship between the crystal size distribution and DSC melting traces, and propose a new method to calculate the crystal size number distribution. A detailed calculation method is proposed according to the recently proposed generalized melting temperature equation (Kamal et al., 2002). The new method is used to calculate the number average crystal sizes (lamellar thickness) for several linear low-density polyethylenes (LLDPEs) with different types of molecular structures, and one linear polyethylene. The calculated average crystal sizes are compared with small angle x-ray scattering (SAXS) results. The crystal size polydispersity is evaluated from the number average crystal size and the weight average crystal size. Theoretical AnalysisThe DSC melting curves indicate the heat fl ow as a function of time. In non-isothermal experiments, they can be converted into temperature functions. Furthermore, they can be converted into crystal size (lamellar thickness) functions, if no other factors are involved, such as the crystal multiphase transitions and crystal melting-recrystallization-remelting (MRR). * Author to whom correspondence may be addressed. E-mail address: musa.kamal@mcgill.caMelting curves, obtained by differential scanning calorimetry, are used to estimate crystal size distributions. The proposed theoretical analysis is applied to different types of polyethylene, including highdensity polyethylene (HDPE), metallocene catalyzed linear low-density polyethylenes (m-LLDPE), ble...

he melting temperature (J,,,) is one of the most important properties of semi-crystalline polymers, especially in the study of crystallization T and melting processes. In polymer processing, the energy requirements of the process, the behaviour of the material during processing, and the morphology of the final product are strongly influenced by the melting and crystallization of the material. Therefore, it is useful to obtain dependable relationships or equations for the estimation of the melting temperatures of homopolymers and copolymers.One of the commonly used melting parameters, Tm0, refers to the thermodynamic equilibrium melting temperature of the polymer crystal with crystal stems containing an infinite number of structural units (Mandelkern, 1964). Obviously, this is a theoretical property, since it is not possible to achieve a polymer with infinite molecular weight. Usually, Tm0 is estimated for homopolymers by extrapolation according to established relationships, such as the Cibbs-Thomson equation (Hoffman et al., 1976) and the Hoffman-Weeks equation (Hoffman and Weeks, 1962), or by extrapolation according to the melting properties of a series of small molecules (Flory and Vrij, 1963). In this paper, we refer to 7, " for homopolymers as T,,,"' ,-. For copolymers, Flory (1 955) proposed theoretical calculation methods by consideration of crystals that excluded comonomers from the crystals. The treatment is based on analysis of the depression of Tm0 (JmH,-) by the incorporation of the comonomers. Therefore, in this work, we shall refer to the melting temperature of a copolymer calculated following Flory's equation asIn real terms, one should consider finite dimensions of the crystal. Real crystal dimensions are finite, because of the finite molecular weights and the exclusion of branch units. So, T, , , " does not have real physical meaning in real crystals. It is more practical to consider the melting temperature of the crystal with crystal stems containing the maximum possible size or number of structural units (n.), defined here as T,,,"' .For homopolymers and copolymers with included comonomers, T, " ' is the melting temperature of the molecular crystal. However, it is also very difficult to form molecular crystals, especially in high molecular weight polymers, because of limitations associated with chain flexibility, flow viscosity, etc. Therefore, this kind of melting temperature is difficult to achieve experimentally. For copolymers with excluded comonomers, because the excluded comonomer units become the lattice ends, the r,,,+.

In this study the crystallization behavior of linear low‐density polyethylenes (LLDPEs) (ethylene‐α‐olefin copolymers) was studied by polarized light microscopy. A modified Hoffman‐Lauritzen (MHL) expression is proposed whereby the equilibrium melting temperature, T mH,∞ (T m0), is replaced with the melting temperature of the crystal stem is replaced with the maximum possible stem length, T mC,n*. It successfully describes the crystalline spherulitic growth kinetics for both homogeneous and heterogeneous LLDPEs. In addition to regimes III and II, another regime (IM) was found in the high crystallization temperature range. Linear growth behavior of crystalline spherulites was observed in regime III, and nonlinear growth behavior was found in regimes II and IM. The basal surface free energy can be estimated from the short chain branching polydispersity (SCBP) for LLDPEs with excluded comonomers. Polym. Eng. Sci. 45:74–83, 2005. © 2004 Society of Plastics Engineers.

The Linkam shearing cell was used in conjunction with a polarized light microscope to study the isothermal crystallization process of linear low‐density polyethylene resins under simple shear flow. The growth of spherulitic morphology was observed under slow shear rates (less than 1 s−1). The crystalline spherulitic growth rate increases, as the shear rate increases. This has been attributed to the decrease of the activation diffusion energy. A relationship between the activation diffusion energy and the shear rate is proposed, under the experimental conditions employed. The modified Hoffman–Lauritzen equation successfully describes spherulitic crystallization kinetics under shear conditions, when the appropriate value of activation diffusion energy is employed. POLYM. ENG. SCI. 46:1468–1475, 2006. © 2006 Society of Plastics Engineers

The crystallization behavior of homogeneous and heterogeneous linear low‐density polyethylenes (LLDPE) was investigated by evaluating the characteristics of melting traces obtained by differential scanning calorimetry (DSC). Based on the isothermal experimental results, the concept of the effective nucleation induction time is suggested. In the initial crystallization stage, the Avrami equation in conjunction with the effective induction time can be used to successfully describe the overall crystallization kinetics. Avrami exponents 2, 1.5, and 1 were found to apply in regimes III, II, and IM, respectively, as identified by the modified Hoffman‐Lauritzen (MHL) equation. The kinetic parameters estimated from evaluating the linear crystallization behavior during spherulitic growth experiments using polarized light microscopy (PLM) are in agreement with the overall crystallization kinetic parameters obtained from DSC experiments. POLYM. ENG. SCI., 45:1140–1151, 2005. © 2005 Society of Plastics Engineers