Crystallization in statistical copolymers

In what follows, we use simple mean-field theories to predict polymer phase diagrams and then use numerical simulations to study the kinetics of polymer crystallization behaviors and the morphologies of the resulting polymer crystals. More specifically, in the molecular driving forces for the crystallization of statistical copolymers, the distinction of comonomer sequences from monomer sequences can be represented by the absence (presence) of parallel attractions. We also devote considerable attention to the study of the free-energy landscape of single-chain homopolymer crystallites. For readers interested in the computational techniques that we used, we provide a detailed description in the Appendix. ... 

In semicrystalline block copolymers, the crystallization behavior is often more complex than that observed in statistical copolymers because the solid-state morphology adopted by block copolymers can be driven either by block incompatibility or by crystallization of one or more blocks [5-8]. In this chapter, we will cover only block copolymers with homogeneous or weakly segregated melts, such that crystallization is always the dominant factor in determining solid-state morphology. Crystallization of block copolymers from strongly segregated melts is covered in Chapter 12. Furthermore, the... 

Statistical copolymers refer to a class of copolymers in which the distribution of the monomer counits follows Markovian statistics [1,2]. In these polymeric materials, since the different chemical units are joined at random, the resulting polymer chains would be expected to encounter difficulties in packing into crystaUine structures with long-range order however, numerous experiments have shown that crystallites can form in statistical copolymers under suitable conditions [2], In this section, we will discuss the effects of counit incorporation on the solid-state structure and the crystallization kinetics in statistical copolymers. A number of thermodynamic models, which have been proposed to describe the equilibrium crystallization/melting behavior in copolymers, vill also be highlighted, and their applicability to describing experimental observations will be discussed. 

In many ways, crystallization of statistical copolymers resembles that of homopolymers [65, 68-70]. For example, hydrogenated polybutadienes of different branch content and molecular weight show sigmoidal crystallization isotherms, similar to those seen in polyethylene homopolymer crystallization. In addition, at a given isothermal crystallization temperature, the degree of copolymer crystallinity drops above a critical molecular weight, also qualitatively similar to polyethylene homopolymer [69]. 

Thus far, we have discussed a number of key experimental observations regarding the effects of counit incorporation on the solid-state structure and the crystallization kinetics in statistical copolymers. In order to better quantify these experimental observations, various thermodynamic models have been proposed. Rory s model, as outlined in Section 11.2.1, correctly describes the equilibrium melting behavior of copolymers in the limit of complete comonomer exclusion however, it is often found to be inadequate at predicting experimentally accessible copolymer melting temperatures [11-14]. An alternative was proposed by Baur [91], where each polymer sequence is treated as a separate molecule with an average sequence length in the melt given by [91] ... 

Besides its effects on morphology, comonomer sequence distribution also affects copolymer crystallization kinetics. In statistical copolymers, due to the broad distribution of crystaUizable sequence lengths, bimodal melting endotherms are typically observed. In block copolymers, the dynamics of crystallization have features characteristic of both homopolymer crystallization and microphase separation in amorphous block copolymers. In addition, the presence of order in the melt, even if the segregation strength is weak, hinders the development of the equihbrium spacing in the block copolymer solid-state structure. 

Statistical copolymers of the types described in Chapter 8 tend to have broader glass transition regions than homopolymers. The two comonomers usually do not fit into a common crystal lattice and the melting points of copolymers will be lower and their melting ranges will be broader, if they crystallize at all. Branched and linear polyethylene provide a case in point since the branched polymer can be regarded as a copolymer of ethylene and higher 1-olefins. 

Inclusion compounds allow the realization of copolymerization in the crystal state (1-6). This is a further difference with respect to typical solid state reactions. Both block- and statistical copolymers can be obtained the former involves a two-step process, with subsequent inclusion and polymerization of two different monomers (21) the latter requires the simultaneous inclusion of two guests. This phenomenon has a much wider occurrence than thought at first, especially when a not very selective host such as PHTP is used. Research with this host started with mixtures of 2-methylpentadiene and 4-methylpentadiene, two almost exactly superimposable molecules (22), but was successfully extended to very dissimilar monomers, such as butadiene and 2,3-dimethylbutadiene. 

Moreover, some polymers cannot be crystallized even in principle. Indeed, crystallization may only appear if there is long-scale order in the molecules positions (as in Figure 4.1 b). However, say, for a statistical copolymer whose chains consist of two types of units, A and B, long-scale order is impossible. (This is simply because the sequences of A and B along the chains are totally random.) Such copolymers can never crystalhze on cooling. 

The principles of polymer fractionation by solubility or crystallization in solution have been extensively reviewed on the basis of Hory-Huggins statistical thermodynamic treatment [58,59], which accounts for melting point depression by the presence of solvents. For random copolymers the classical Flory equation [60] applies ... 

This kind of sequence defect occurs in the statistical copolymers, where the species of monomers can crystallize. On the backbone of polyethylene chains, the short branches can be regarded as the non-crystallizable comonomers. In high-density polyethylene (HOPE), the branching probability is about 3 branches/1,000 backbone carbon atoms, and its crystallinity can reach levels as high as 90 % while in low-density polyethylene (LDPE), the branching probability is about 30 branches/ 1,000 backbone carbon atoms, and its crystallinity reaches only 50 %. The most common industry product is actually linear low-density polyethylene (LLDPE), and its branching probability is determined by the copolymerization process of CH2 = CH2 and CH2 = CHR (R means side alkane groups for short branches). 

Flory PJ (1954) Theory of crystallization in copolymers. Trans Faraday Soc 51 848-857 Flory PJ (1956) Statistical thermodynamics of semi-fiexible chain molecules. Proc R Soc London A234 60-73... 

Polymers that cannot crystallize usually have some irregularity in their structure. Examples include the atactic vinyl polymers and statistical copolymers. 

Flory PJ (1941) Thermodynamics of high polymer solutions. J Chem Phys 9(8) 660 Flory PJ (1942) Thermodynamics of high polymer solutions. J Chem Phys 10(1) 51-61 Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, Ithaca Flory PJ (1955) Theory of crystallization in copolymers. Trans Faraday Soc 51 848-857 Flory PJ (1956) Statistical thermodynamics of semi-flexible chain molecules. Proc R Soc Lond A Math Phys Sci 234(1196) 60-73... 

It is useful to divide the polymers into two main classes the fully amorphous and the semicrystal-line. The fully amorphous polymers show no sharp, crystalline Bragg reflections in the X-ray diffracto-grams taken at any temperature. The reason why these polymers are unable to crystallize is commonly their irregular chain structure. Atactic polymers, statistical copolymers and highly branched polymers belong to this class of polymers (Chapter 5). 

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