Vertex functions in the Landau theory of phase transitions in melts of Markovian multiblock copolymers

Aliev, M. A.; Кучанов, С.И.

doi:10.1016/j.physa.2007.09.024

Cited by 8 publications

(23 citation statements)

References 18 publications

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“…If it is three dimensional, then k = . In particular, for a lamellar mesophase (d = 1) and a hexagonal mesophase (d = 2), respectively, we have (10) Here and below, L denotes the linear dimension of the unit cell of the lattice, which characterizes the symme try of a given mesophase. If it is lamellar, L is the length of a straight line segment; if the mesophase is hexagonal, L is the length of a regular hexagon.…”

Section: Spectral Methodsmentioning

confidence: 99%

“…In this case, free energy func tional linearly depends on the volume fractions of the coexisting phases, and the free energy of each of them is described through a single replica functional in a conventional three dimensional Euclidean space [7,11,18]. The minimization of this functional, the expression for which was first given in [19,20], allows solution of the following important problems of ther modynamics of block copolymers: (i) finding a variety 10 11 12 20 21 22…”

Section: Model and Formulation Of The Problemmentioning

confidence: 99%

“…There are a few publications concerned with the implementation of the procedure for determining the vertex functions that characterize the free energy of melts of statistical and block copolymers [7][8][9][10][11][12] as well as the construction of phase diagrams of melts of these copolymers in terms of the weak segregation approach [13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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Features of the phase behavior of block copolymers related to their polydispersity

2012

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To calculate cloud points of a melt of block copolymers, at which its microphase separation occurs, for the first time in the theory of polymers, a spectral method is used in a model that takes into account the compressibility of the melt and the polydispersity of its macromolecules. A particular case of this calculation for a copolymer whose macromolecules consist of two blocks is described, and the polydispersity of each of them is defined through the exponential Flory distribution.

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Section: Spectral Methodsmentioning

confidence: 99%

Section: Model and Formulation Of The Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Features of the phase behavior of block copolymers related to their polydispersity

2012

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“…(2.12) and (2.13) are, respectively, the persistence length l p and the contour length l c of the real polymer, i.e., l p < b < R e < l c . 4 In terms of the model segments b s := R s − R s−1 , s = 1, . .…”

Section: Polymer Chain Connectivitymentioning

confidence: 99%

“…In order to calculate the free-energy density in macroscopic phase separation, we have to derive a real-space version of the free-energy functional which allows for a correct ansatz of multiple homogeneous phases before taking the thermodynamic limit. 4 Fortunately, by redefining our set of variables for the cases in which eq. (3.32) indicates macroscopic phase separation, we can derive for the free energy of coexisting homogeneous phases a closed expression that is not limited to small order-parameter amplitudes or weak A-B segregation.…”

Section: Macroscopic Phase Separationmentioning

confidence: 99%

Multiphase coexistence with sequence fractionation in random block copolymers

Heydt¹

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Random binary block copolymers emerge from linking permanently and at random prepolymer blocks of two different chemical species A and B. The competitive interplay of conformational entropy, connectivity within one polymer, temperature-dependent incompatibility between A and B, and incompressibility gives rise to a complex phase behavior with a variety of possible morphologies of A-and B-rich domains. Technical applications of the self-organized structures in block copolymers include nanoscale templates and medical drug delivery via copolymer micelles.For random Q-block copolymers, this work addresses theoretically the conjectured coexistence of macroscopic phase separation and a structured phase of microscopic A-and B-rich domains. Sequence fractionation according to the copolymers' internal structure promotes the coexistence of phases with different morphologies in equilibrium, as is revealed by a theory with explicit account for the exchange of individual sequences. In our semi-microscopic model, one block comprises M identical segments. The Markovian block-type sequence distribution is characterized by the type correlation λ of adjacent blocks and the global A content. Our focus is on block copolymer distributions with A B exchange symmetry, for which phase transitions from the disordered state are continuous within mean-field theory. Upon increasing the incompatibility χ (by decreasing temperature) in the disordered state, we observe the formation of the known global, ordered phases: for λ > λ c , two coexisting macroscopic A-and B-rich phases, and for λ < λ c , a microstructured (lamellar) phase with nonzero wave number, k(λ). In addition, we encounter a fourth region in the λ-χ plane where these three phases coexist with different, for Q ≥ 3 non-Markovian, sequence distributions. The three-phase region is reached, either from the macroscopic phases via a third lamellar phase that is rich in alternating sequences, or starting from the lamellar state, via two additional homogeneous, homopolymer-enriched phases; in both cases, the incipient phases have zero volume fraction. The four regions of the phase diagram meet at a multicritical point (λ c , χ c ), at which A-B segregation vanishes. Since our analytical method assumes weak segregation for the lamellar phase, it proves reliable particularly in the vicinity of (λ c , χ c ). For random triblock copolymers, Q = 3, we find that both the character of this point and the critical exponent of the segregation amplitude change substantially with the number M of segments per block: The lamellar wave number vanishes continuously on approach to (λ c , χ c ) only for M < 7. The results for Q = 3 in the continuous-chain limit M → ∞ are compared to numerical self-consistent field theory (SCFT), which is accurate at larger segregation.

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Microphase separation in polydisperse miktoarm star copolymers

Aliev

Kuzminyh

2011

Physica A: Statistical Mechanics and its Applications

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Vertex functions in the Landau theory of phase transitions in melts of Markovian multiblock copolymers

Cited by 8 publications

References 18 publications

Features of the phase behavior of block copolymers related to their polydispersity

Features of the phase behavior of block copolymers related to their polydispersity

Multiphase coexistence with sequence fractionation in random block copolymers

Microphase separation in polydisperse miktoarm star copolymers

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