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Continuous thermodynamics of copolymer systems

  • Margit T. Rätzsch
  • Christian Wohlfarth
Conference paper
Part of the Advances in Polymer Science book series (POLYMER, volume 98)

Abstract

Continuous thermodynamics provides a simple way for the thermodynamic treatment of polydisperse systems. Such systems consist of a very large number of similar species whose composition is described not by the mole fractions of the individual components but by continuous distribution functions. For copolymers, multivariate distribution functions have to be used for describing the dependence of thermodynamic behavior on molar mass, chemical composition, sequence length, branching, etc.

In this paper, we review continuous thermodynamics as applied to copolymer systems. Special attention is focused on liquid-liquid equilibria and thermodynamic stability. Equilibria in solutions of random copolymers, blends of random copolymers with homo- or copolymers, and also the high pressure phase equilibrium in the mixture of copoly(ethene vinylacetate) with its monomers are also discussed. A special examination of polydispersity effects in solutions with block copolymers is made. Thus, the paper reviews in a comprehensive way how to build up continuous thermodynamics with multivariate distribution functions and how to derive relations necessary for solving special problems. Some short remarks on possible future prospects will round up the paper.

Keywords

Block Copolymer Random Copolymer Coexistence Curve Segment Number Continuous Distribution Function 
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 Symbols

a; b; c; d;

Fitting parameters in \(\bar \bar G\)E-relations (eqs. (78), (100), (134), (150))

AB; BB

Abbreviations (Eqs. (127), (170))

C1

Abbreviation (Eq. (118))

dm

Periodicity distance between microdomains in block copolymer systems (Eq. (190))

D

Number of polydisperse copolymers (distributions)

D1

Abbreviation (Eq. (97))

f0; f1; f2

Fitting parameters in χ-expression (Eq. (78))

f(M1, M2, ..., Mt)

Any continuous function of variables M1, M2, ..., Mt (Eq. (21))

\(\bar \bar f\)A; \(\bar \bar f\)B(M1, M2, ..., Mt);\(\bar \bar f\)B(rB, YB)

Segment-molar activity coefficients (Eqs. (22), (25), (53), (62), (64))

ΔFMST

Difference of free energies between a random and a mesophase (Eq. (186))

FU

Abbreviation (Eqs. (92), (93), (168))

gαβ; gαγ; ...

Interaction parameters in \(\bar \bar G\)E-model (Eqs. (150)–(155))

g

Interaction parameter function (Eqs. (150), (155))

Δgres

Residual part of Gibbs free energy of mixing (Eq. (72))

gijk

Coefficients of a power series of ΔFMST (Eq. (188))

\(\bar \bar G\)

Segment-molar Gibbs free energy (Eqs. (29), (30))

\(\bar \bar G\)E

Segment-molar excess Gibbs free energy (Eqs. (30), (78), (150))

I

Number of solvents A (of individual components)

I1, I2

Integrals (Eqs. (128))

J

Number of polydisperse copolymers in blends

k

Boltzmann's constant

kB

Parameter of the generalized Stockmayer distribution (Eq. (85))

kαβ; kβα; kαα; kββ

Rate constants of the basic propagation reactions of copolymerization (Eq. (86))

{LB}; {LK}

Set of additional identification variables (Eqs. (65), (172)–(179))

L(ψB)

Concentration function in \(\bar \bar G\)E-expression (Eqs. (101), (102) and (150), (151))

M1, M2, ..., Mt

Variables of the distribution function (Eq. (3))

\(\widetilde{M_B^n };\widetilde{M_K^n }\)

n-th moments of the distribution functions of B or K with respect to MB or MK, respectively (Eqs. (68), (176))

n; \(\bar \bar n\)

Total amount of substance or segments, respectively (Eq. (1))

\(\bar \bar n\)A

Amount of segments of solvent A (Eq. (4))

{\(\bar \bar n\)A}

= \(\bar \bar n\)A′, \(\bar \bar n\)A″, ..., \(\bar \bar n\)A(1)(Eq. (10))

\(\bar \bar n\)B

Overall amount of segments of all species of copolymer B (Eq. (3))

NAv

Avogadro's number

N*

Contact pair number (Eq. (72))

p, q

Auxiliary quantities (Eqs. (130))

P

Pressure

q11BB, q11CC, ...

Elements of the matrix Qs (Eq. (163))

Qk (\(\vec x,t;\vec x_0\))

Configuration partition function of a Gaussian chain k

\(\tilde Q\)k

Functional of Qk(\(\vec x,t;\vec x_0\)) (Eq. (186))

Qs

Stability matrix (Eq. (162))

r

Segment number (Eq. (1))

\(\bar r\)

Number average segment number of the total phase considered (Eq. (27))

rA

Segment number of solvent A (Eq. (1))

rB

Segment number of a copolymer species (Eq. (1))

r0, r0

Lower and upper integration limit with respect to rB (Eq. (32))

\(\bar r\)BB

Number average segment number of copolymer B (Eq. (28))

rmB

Segment number of a certain monomer unit of a copolymer B (Eq. (24))

\(\widetilde{r_B^n }\)

n-th moment of the distribution function with respect to rB (Eq. (34))

r11K, \(\hat r\)11K, ...

Elements of the matrix RK and the inverse matrix \(\hat R\)K, respectively, (Eqs. (163)–(167))

R

Universal gas constant

RK, \(\hat R\)K

Matrix and inverse matrix

Sα, Sβ, Sγ, Sδ

Surface contact parameters (Eq. (155))

T

Temperature

V

Molar volume

u, u, uαβ

Interaction energies (Eq. (72))

WB(M1, M2, ..., Mt)

Extensive distribution function (Eq. (3))

{WB}

= WB′, WB″, ..., WB(D) (Eq. (11))

WB(M1, M2, ..., Mt)

Intensive segment-molar distribution function (Eq. (7))

\(\vec x,\vec y\)

Vectors

X

Abbreviation solution of Eq. (129)

YB, YC

Segment-fractions of α-units or γ-units of a molecule of copolymers B or C, respectively (Eq. (31))

\(\widetilde{Y_B^m }\)

m-th moment of the distribution function with respect to YB (Eq. (34))

\(\bar Y\)B(i)

i-th moment of the non-normalized distribution function with respect to YB (Eq. (164))

z

Coordination number (Eqs. (74)–(78))

z

Extensive property (Eq. (12))

\(\bar \bar Z\)

Segment-molar property (Eqs. (15), (16))

\(\bar \bar Z\)A; \(\bar \bar Z\)B(M1, M2, ..., Mt)

Partial segment-molar properties (Eqs. (18), (19))

Greek Symbols

α; β; γ; δ

Copolymer units (Eqs. (31), (153))

αi, βi

Auxiliary quantities (Eqs. (132), (133))

Γ

Gamma-function (Eq. (84))

δ

Differential (Eq. (13))

δ(M′ - M)

Dirac-function of Variable M (Eqs. (20), (21))

Δ

Difference

εB

Parameter of the Stockmayer distribution (Eq. (84))

ϰ1, ϰ2

Parameters of the model distribution function (Eq. (124))

η1, η2

Average surface parameters of copolymers B and C, respectively, (Eq. (153))

λ

Weight factor (Eq. (134))

\(\bar \bar \mu\)A, \(\bar \bar \mu\)B(M1, M2, ..., Mt)

Segment-molar chemical potentials (Eqs. (22), (25))

\(\bar \bar \mu\)A*, \(\bar \bar \mu\)B, 0*(M1, M2, ..., Mt)

Segment-molar reference chemical potentials (Eqs. (22), (25), (26))

vb, vc

Fitting parameters in χ-relations (Eq. (80))

ϱA; ϱB(rB, YB); ϱK(rK, YK, {LK})

Abbreviations (Eqs. (36)–(39], (138), (173))

ϱ0

Reference number density (Eq. (186))

φ

Relative amount of segments of phase II (Eq. (44))

χ

Huggins′ χ-parameter (Eqs. (78), (99), (134), (150))

ψA

Segment-fraction of solvent A (Eq. (5))

ψB, ψK

Segment-fraction of all species of copolymer B or K, respectively (Eqs. (6), (179))

ψi\((\vec x);\bar \psi _i (\vec x)\)

Local and residual local segment (volume) fraction of species i at point x (Eq. (186))

ωi(\(\vec x\))

Mean field potential acting on a segment of species i at point x (Eq. (186))

Subscripts

A

Solvent

A′, ..., A(I)

Several solvents (individual components)

A1, A2

Two solvents

B

Polydisperse copolymer

B′, ..., B(D)

Several polydisperse copolymers

B, C, (..., J)

Two (or more) polydisperse copolymers in blends

crit

Critical property

i; j; k; m; n; ...

Indices in sums

α; β; γ; δ

Copolymer units

Superscripts

I, II

Coexisting phases I, II

F

Feed phase

E

Excess

FT

Fourier transform

MST

Mesophase separation transition

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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Margit T. Rätzsch
    • 1
  • Christian Wohlfarth
    • 1
  1. 1.Chemistry Department“Carl Schorlemmer” Technical UniversityMerseburgFRG

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