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Statistical theory of polymeric lyotropic liquid crystals

  • Alexandr Yu. Grosberg
  • Alexei R. Khokhlov
Conference paper
Part of the Advances in Polymer Science book series (POLYMER, volume 41)

Abstract

This article deals with some topics of the statistical physics of liquid-crystalline phase in the solutions of stiff chain macromolecules. These topics include: the problem of the phase diagram for the liquid-crystalline transition in the solutions of completely stiff macromolecules (rigid rods); conditions of formation of the liquid-crystalline phase in the solutions of semiflexible macromolecules; possibility of the intramolecular liquid-crystalline ordering in semiflexible macromolecules; structure of intramolecular liquid crystals and dependence of the properties of the liquid-crystalline phase on the microstructure of the polymer chain.

Keywords

Triple Point Virial Coefficient Anisotropic Phase Triple Point Temperature Small Globule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Main Symbols

N

total number of particles in the system, i. e. in Sects. 2, 3 — total number of rods (segments) in the solution, in Sects. 4, 5 — total number of rods (segments) in the macromolecule

L

contour length of a macromolecule

d

width of a polymer chain

statistical segment length (in Sect. 5.5 — length of a rod-like side group)

p

assymmetry parameter, i. e. in Sect. 2 p=L/d, in Sect. 3 p=1/d

γ

angle between the interacting rods (segments)

f(u)

orientational distribution function for the rods (segments)

element of a spatial angle

c

concentration of rods (segments)

ϑ

polymer volume fraction in the solution

ϑ(i) and ϑ(a)

polymer volume fractions in the isotropic and anisotropic phases coexisting at equilibrium

T

absolute temperature in energetic units

F

free energy

θ

theta temperature

T3

temperature corresponding to the triple point of the phase diagram

(−ε)

energy of interaction of two perpendicular rods (segments) in contact

Z

lattice co-ordination number in lattice models

x=(Z−2)ε/2T

Flory interaction parameter

B

second virial coefficient

C

third virial coefficient

ξ (in Sect. 2), λ

see Eq. (2.21)

v

excluded volume of a monomer

a

characteristic distance between the ends of a flexible filament between the monomers

z=vN1/2/a3

excluded volume parameter of the coil

ϑ0

polymer volume fraction inside the globule

s

parameter of liquid-crystalline order

R, r

radii of a toroidal globule (Fig. 12)

ξ (in Sect. 5)

set of parameters defining the state of a monomer

n(ξ)

density of monomers in the state ξ

ψ(ξ) and ψ+(ξ)

distribution functions for the end monomers of the macromolecule

g(ξ, ξ′)

conditional probability that the monomer is in the state ξ′ provided the previous one is in the state ξ

ĝ

integral operator with the kernel g(ξ, ξ′) (see Eq. (5.2a))

S{n}

conformation entropy of a globule with the given density distribution n(ξ).

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Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Alexandr Yu. Grosberg
    • 1
  • Alexei R. Khokhlov
    • 2
  1. 1.Institute of Chemical PhysicsUSSR Academy of SciencesMoscowUSSR
  2. 2.International Centre for Theoretical PhysicsTriesteItaly

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