Copolymer Blend Theory

For simplicity, the monomeric units of AA, styrene and EO are denoted as a, b, and c instead of using their full names. The free energy of mixing for a binary mixture of a homopolymer and a copolymer, cNi / (dfbl )N2, is given by [12-14] 

However, the simple binary interaction model is inadequate to study the sequence effects owing to its assumption of a random distribution. Assuming that the interaction energy parameters of all a-a and b-b pairs are equivalent and equal to zero and that all a-b interactions are equivalent to the average interaction parameter %ab, Balazs et al. [16] expressed xtotas the sum of the contribution of the composition Xcomp an(i the sequence distribution Xdistas follows  

A negative Aya implies that aaa-c interactions are energetically more favorable than any other type of a-c interactions, and a similar comment is applicable to Axa- The parameters 0 and 6 are introduced to specify the sequence distribution and the composition, respectively  

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Karasz et al. [20] considered AB-AB type blends. Per the first-order random copolymer blend theory as described in Section 3.9 the interaction parameter of the blend can be written as... 

Based on the pioneering work of Molau [64], it is evident that phase separation can occur in blends of two or more copolymers produced from the same monomers when the composition difference between the blend components exceeds some critical value. The mean field theory for random copolymer-copolymers blends has been applied to ES-ES blends differing in styrene content to determine the miscibility behavior of blends [65,66]. On the basis of the solubility parameter difference between PS and PE, it was predicted that the critical comonomer difference in styrene content at which phase separation occurs is about 10 wt% S for ESI with molecular weight around 105. DMS plots for ES73 and ES66 copolymers and their 1 1 blend are presented in Figure 26.8. 

Vilgis TA, Noolandi J. Theory of homopolymer—block copolymer blends the search for a universal compatibilizer. Macromolecules 1990 23 2941-2947. 

The crystalline bicomponent and biconstituent blends considered above may also be compared to the polyallomer block copolymers (Section 6.8), which consist of two crystalline components linked together by chemical bonds rather than by mechanical mixing. In all cases, the constituents are independently crystallizable. Melt-blending theory is considered in Section 9.6. 

A special problem in the treatment of the denuxing equihbrium is the calculation of spinodal curve and critical point. If the stabDity theory in the framework of continuous thermodynamics [45 -47] is applied to copolymer blends, it is shown that the spinodal curve is the solution to the equation... 

However, many problems and applications still remain unattacked when this review is prepared. Further progress will take place when multivariate distribution functions become available by experiments of higher accuracy than now. More exact and sophisticated G -models have to be developed for the application of continuous thermodynamics to copolymer systems. New insights into the delicate phase behavior of copolymer systems would be gained by further development of the stability theory of continuous thermodynamics [45-47,75]. The polymer fractionation theory by continuous thermodynamics should be extended from homopolymers [100] to copolymers. In short, much remains to be done in the field of copolymer blends and systems containing block copolymers. 

G. Floudas, N. Hadjichristidis, M. Stamm, A.E. Likhtman, A.N. Semenov, Microphase separation in block copolymer/homopolymer blends theory and experiment. Journal of Chemical Physics 106 (1997) 3318-3328. 

An important application of block copolymers is as compatibilizers of otherwise immiscible homopolymers. There are a number of useful reviews of work in this area (179-182). The morphology of blends of polymers with block copolymer and theories for this have been reviewed (1). The influence of added homopolymer... 

A miscibility window was identified in the temperature-composition plane. The miscibility windows of polyphenylene oxide/orthochlorostyrene/parachlrostyrene (PPO/ oClS-pClS) and polyphenylene oxide/orthoflurostyrene-parafluorostyrene (PPO/oFS-pFS) were compared with each other. The maxima in the miscibility window of PPO/ oClS-pClS were found at the center of the composition axis, and the maxima in the PPO/oFS-pFS system were found skewed to the o-FS-rich side. The miscibility window was not observed for the PPO/o-bromostyrene and / -bromostyrene copolymer blend system. Kambour et al. [1] formulated a Flory-Huggins type theory for mixtures of homopolymers and random copolymers. They argued that such a system can be miscible for a suitable choice of the copolymer composition, without the presence of any specific interaction, because of a so-called repulsion between the two different monomers comprising the copolymer. 

It can be noted that the observed miscible polymer systems may be due to intramolecular repulsion rather than intermolecular attraction as originally hypothesized. The miscible blends of random copolymer (AB) and a homopolymer (C) in AB/C type blends were found miscible over a range of compositions of comonomer 1 in copolymer AB. This indicates a possible effect of the chain sequence distribution on the miscibility. Even in cases where the homopolymers were not miscible, the copolymer and homopolymer were found to be miscible. The mean-field theory of random copolymer blends is an important development in the thermodynamics of polymer miscibility. The compositional window of miscibility has been found for systems where A, B, and C were found to be immiscible as homopolymers, but were found to be miscible as A-B copolymer and C homopolymer. 

What is the significance of the mean-field theory of random copolymer blends in the thermodynamics of polymer miscibility ... 

Our understanding of the equilibrium microphase behavior of block copolymers, and certain aspects of block copolymer blends, is well advanced. The NSCFT provides an excellent description of the equilibrium phases and domains of block copolymers of various architectures and species, but with one important exception the region near the order-disorder transition for nearly symmetric molecules. It appears that understanding this region will require a theory that includes both the high order effects incorporated within the NSCFT, and fluctuation effects. The theory can also provide detailed descriptions of the amorphous regions of semicrystallizable copolymers. 

Horst, 1995, Calculation of phase-diagrams not requiring the derivatives of the gibbs energy demonstrated for a mixture of 2 homopolymers with the corresponding copolymer, Macromol. Theory Simid., Vol. 4, No. 3, PP. 449-458 Horst Wolf, 1992, Calculation of the phase-separation behavior of sheared polymer blends, Macromolectdes, Vol. 25, No. 20, PP. 5291-52%... 

SAT Sato, T., Tohyama, M., Suzuki, M., Shiomi, T., and Imai, K., Application of equation-of-state theory to random copolymer blends with upper critical solution temperature type miscibility, M3crowo/ecM/es, 29, 8231, 1996. 

Theories of the Intetfacial Behavior in Homopolymer/ Homopolymer/Copolymer Blends... 

Whitmore MD, Noolandi J. Theory of crystaUizable block copolymer blends. Macromolecules 1988 21 1482-1496. 

Lattice Cluster Theory gives in addition an entropic term which for the PB copolymer blends reads... 

An additional dependence on the copolymer compositions x and y beyond that predicted by simple random copolymer FH theory appears in Eq. 24b through the front conversion factor C/S1S2 as well as through factors of 51/52 and 52/51. The main novel feature of Eq. 24b, however, lies in the presence of the temperature-independent portion of xsans. This term represents the influence of monomer structural asymmetry on the nonrandom chain packing and coincides with the leading contribution of the LCT for binary blends in the incompressible, high molecular weight, athermal, fully flexible chain limit. 

As illustrated in the next subsections, the BLCT has successfully been applied to a wide variety of copolymer blends whose thermodynamics cannot be described by random copolymer FH theory. While both theories share the inability to distinguish between random, diblock, or alternating copolymers of the same composition, the sensitivity of computed thermodynamic properties to the detailed monomer molecular structure of the blend components makes the BLCT a much more powerful theoretical tool than random copolymer FH theory. The relative simplicity of Eq. 24b and the readily computed entropic factors n and ri render the BLCT useful for interpreting thermodynamic data and for generating new predictions. 

The observation by Delfohe et al. [96] that the miscibility of norbornene-co-ethylene (NxEi-x/NyEi- ) binary copolymer blends improves significantly when both components are rich in norbornene, i.e., when x > 1/2 andy > 1/2, stands in sharp contrast to the predictions (Eq. 27) of random copolymer FH theory [88], which imphes that the blends are miscible when the composition difference x-y is less than a critical constant value. The general theory of Sect. 8.1 that has been used to explain this observation is described elsewhere in some detail [96]. Here we summarize the main results of this theoretical analysis to illustrate some computational aspects of the BECT. 

When applied to systems containing statistical copolymers, the SLCT loses its enormous analytical and physical simplicity due to the greater complexity of these systems compared to binary homopolymer blends. The lack of mathematical simplicity in describing copolymer blends arises, in part, from the dependence of the free energy on the monomer sequence. Therefore, a further approximation is introduced into the SLCT to generate a theoretical approach that is simple and easy to use but is devoid of a serious deficiency of the extensions of FH theory to random copolymer systems, namely, the neg-... 

As mentioned earlier, the SLCT has been derived for Ax i-x/CyD -y copolymer systems [93], but the theory has not yet been appfied to analyze experimental data. In contrast to the BLCT, the SLCT predicts that the energetic portion x of the xsANS parameter for copolymer blends depends on the blend composition and is sensitive to the monomer sequence distribu-... 

Banaszak M, Whitmore MD. Self-consistent theory of block copolymer blends - selective solvent. Macromolecules 1992 25 3406-12. 

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