Block Copolymer Theory. 6. Cylindrical Domains

Helfand, Eugene; Wasserman, Zelda R.

doi:10.1021/ma60076a045

Cited by 261 publications

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“…The UCA is used to simplify the calculation for C and S by approximating their Wigner-Seitz unit cells by a circular cylinder and a sphere, respectively. 2 With the resulting rotational symmetry, all quantities will depend only on the distance, r, from the center of the unit cell. For convenience, this is expressed in terms of the dimensionless coordinate, F ≡ r/R, where R is the radius of the unit cell.…”

Section: Theorymentioning

confidence: 99%

“…4,18 Here we go one step further than Janert and Schick and approximate the Wigner-Seitz unit cells of the cylindrical and spherical phases by circular and spherical unit cells, respectively. This is the unit-cell approximation (UCA), the use of which was normal practice in earlier SCFT calculations 2,19 and remains so in alternative theories. 7,8 The UCA has been demonstrated to be a good approximation in neat diblock copolymer melts 20 and presumably will remain so for most block copolymer blends.…”

Section: Introductionmentioning

confidence: 99%

“…7,8 The UCA has been demonstrated to be a good approximation in neat diblock copolymer melts 20 and presumably will remain so for most block copolymer blends. Whereas previous UCAbased SCFT calculations solved the self-consistent-field equations in real space, 2,19 we will instead use Bessel function expansions. Not only does this offer significant advantages in controlling numerical accuracy, it also provides an exceptionally efficient algorithm.…”

Section: Introductionmentioning

confidence: 99%

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New Fast SCFT Algorithm Applied to Binary Diblock Copolymer/Homopolymer Blends

Matsen

2003

Macromolecules

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We present an efficient strategy for mapping out the classical phase behavior of block copolymer systems using self-consistent-field theory. With our new algorithm, the complete solution of a classical block copolymer phase can be evaluated typically in a fraction of a second on a single-processor computer, even for highly segregated melts. This is accomplished by implementing the standard unitcell approximation for the cylindrical and spherical phases and solving the resulting equations using a Bessel function expansion. Here the method is used to investigate blends of AB diblock copolymer and A homopolymer, concentrating on the situation where the two molecules are of similar size.

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Section: Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New Fast SCFT Algorithm Applied to Binary Diblock Copolymer/Homopolymer Blends

Matsen

2003

Macromolecules

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“…[1][2][3][4][5][6] However, the microdomain structure disappears completely, giving rise to a disordered state. Such a phenomenon is referred to as the order-disorder transition (ODT).…”

Section: Introductionmentioning

confidence: 99%

Rheology of Polystyrene-block-poly[ethylene-co-(ethylene-propylene)]-block-polystyrene Tri-block Copolymer/Paraffinic Oil Blends - Effect of Oil Molecular Weight on the Order-Disorder Transition -

Takahashi

Nishioka

Ishigami

et al. 2012

J. Soc. Rheol., Jpn

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Blend systems of Polystyrene-block-poly(ethylene-co-(ethylene-propylene)-block-polystyrene(SEEPS) tri-block copolymer and three kinds of paraffinic oil having different molecular weight (MW) were prepared. Styrene content of the SEEPS was 30 wt%. The oils used in this study were Oil-1 (MW=750 g/mole), Oil-2 (540 g/mole), and Oil-3 (410 g/mole). Dynamic viscoelastic properties, the storage (G') and loss modulus (G"), of the blends were measured as a function of temperature (T) and angular frequency (ω). SEEPS/oil = 100/0 and 75/25 (in weight) blends exhibited gel-like behavior over the entire experimental T window below 300 °C. On the contrary, the viscoelastic properties of SEEPS/oil = 50/50 and 25/75 blends depended on the MW of oil. For the SEEPS/Oil-1 = 50/50 and 25/75 blends having high MW oil, the clear order-disorder transition (ODT) was observed showing a sudden change of G' at elevating T and the ODT temperature (T ODT ) was found to shift to lower T with increase in oil content. For the SEEPS/ Oil-2 = 50/50 blend having medium MW oil, the ODT was observed very clearly, but could not be observed for the 25/75 blend. For the SEEPS/Oil-3 blends having low MW oil, the ODT was not observed showing that there is a rather gradual decrease in G' with T. It was found that the viscoelastic properties of SEEPS/oil blends changed with the content and MW of oil.

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“…Meier's theory also stimulated further developments of fundamental statistical mechanical theories of microdomain morphology, such as those by Helfand,I5 Helfand and Wasserman, 16 Leibler,17 Noolandi and Hong," S e m e n o~, '~ Ohta and Kawasaki," Kawasaki and Kawakatq21 Muthukumar and his coworkers," and Matsen et al 23 His theory predicted that the critical block molecular weights required for domain formation are many-fold greater than required for phase separation of a simple mixture of the constituent blocks. This aspect of his study eventually led to another seminal theory of block copolymers, namely, Leibler's Landau-type theory of the phase transition in block copolymer melts-so-called microphase separation transition (MST) or order-disorder transition (ODT).17 Leibler's mean-field theory was further generalized by Fredri~kson-Helfand'~ and Hohenberg-Swift.…”

mentioning

confidence: 96%

Perspective: Comments on “Theory of block copolymers. I. Domain formation in A-B block copolymers,” by D. J. Meier,J. Polym. Sci., C26, 81 (1969)

Hashimoto

1996

J. Polym. Sci. B Polym. Phys.

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At the time this seminal paper by Meier was published, many papers on the colloidal properties of block copolymer solutions had already appeared. The studies involved monomolecular and polymolecular micelles in solution, the possibility of intramolecular phase separation, polymeric oil-in-oil emulsions in systems composed of an A-B block copolymer, a homopolymer and/or B homopolymer in a solvent, and so on.' However, little had been done on the morphology of bulk block copolymers. The state of the art at that stage may be seen in the volumes based on some symposia on block copolymers, e.g., those at the California Institute of Technology in 1967,' in which the Meier's paper was presented, the ACS National Meetings in New York City (1969)3 and Chicago (1970).4 There were quite a number of papers on the thermal and mechanical behavior of block copolymers as thermoplastic elastomer~.'-~ These papers suggested the existence of "microdomain" structures as a consequence of the "microphase separation" between constituent block chains of A and B. However, unequivocal identifications of the microdomain morphologies in bulk block copolymers could not be achieved until the osmium tetraoxide staining method was first introduced to this field by Kato5 in 1967. This technique coupled with transmission electron microscopy on ultrathin sections clearly revealed the phase-separated microdomain structures that were typically a few hundred angstrom in size.'-' Moreover, it showed qualitatively that size and shape of the structures depended on molecular weights of the constituent block chains. These findings naturally led people to recognize that block copolymers can provide "tailormade" supermolecular structures and properties. However, how the domain morphology and its thermodynamic stability depend on molecular and thermodynamic variables was still totally unexplored, though understanding this point is crucial for the rational design of the structures and hence properties of block copolymers. Meier developed the pioneering theory of microdomain morphology in block copolymers and gave a rigorous free energy expression for the microdomain formation." The theory describes criteria for the formation of spherical microdomains and their size in terms of molecular and thermodynamic variables. Further developments of this theory for other microdomain morphologies were reviewed by him." The appearance of his theory was quite timely and stimulated basic experimental studies of (i) the domain size as a function of molecular weight, temperature, and block copolymer concentration when the neutral solvents are used; (ii) the thickness of the interface between two coexisting microdomains; (iii) a long-range order of the microdomains; (iv) morphology as a function of molecular weight ratios of the constituent blocks, and so 0n.12-14Meier's theory also stimulated further developments of fundamental statistical mechanical theories of microdomain morphology, such as those by Helfand,I5 Helfand and Wasserman,16 Leibler,17 Noolandi and Hong," S e m ...

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Block Copolymer Theory. 6. Cylindrical Domains

Cited by 261 publications

References 0 publications

New Fast SCFT Algorithm Applied to Binary Diblock Copolymer/Homopolymer Blends

New Fast SCFT Algorithm Applied to Binary Diblock Copolymer/Homopolymer Blends

Rheology of Polystyrene-block-poly[ethylene-co-(ethylene-propylene)]-block-polystyrene Tri-block Copolymer/Paraffinic Oil Blends - Effect of Oil Molecular Weight on the Order-Disorder Transition -

Perspective: Comments on “Theory of block copolymers. I. Domain formation in A-B block copolymers,” by D. J. Meier,J. Polym. Sci., C26, 81 (1969)

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