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Thermodynamics of polymer solutions under flow: Phase separation and polymer degradation

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Physical Properties of Polymers

Part of the book series: Advances in Polymer Science ((POLYMER,volume 120))

Abstract

The influence of flow on the free energy of polymer solutions is examined from several points of view both on macroscopic and microscopic bases. Application of non-equilibrium chemical potential to the phenomena of flow-induced phase separation and thermodynamically induced polymer degradation is reviewed. Polymer degradation under elongational flow was extensively covered in a recent review in this series. The thermodynamic theory is compared with the dynamical approaches used in the analysis of stability of solutions and it is seen under which conditions the criteria defining the spinodal line under shear (i.e. the limit of the stability region) are the same in both approaches. The thermodynamic analysis may be useful due to its greater simplicity though, in contrast, the details of the phase segregation or homogenisation and the analysis of the dynamical aspects (viscosity, light scattering) are beyond the reach of a strictly thermodynamic method. A short summary of the phenomenology of flow-induced changes in the phase diagram of polymer solutions under flow is given. Perspectives and open problems are pointed out.

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Abbreviations

A:

affinity

A:

matrix of kinetic constants

ai :

activity of the species i

aα :

the droplet radius

b:

length of a segment of the polymer

B:

matrix including the hydrodynamic effects or matrix of kinetic constants

c:

number of moles per unit volume

\(\tilde c\) :

reduced concentration defined as [η]c

d:

slope of the curve η(\(\dot \gamma\)) with a change of sign

D:

curvilinear diffusion coefficient

DT :

effective diffusion coefficient in the shear direction

D 2 :

polymer self-diffusion coefficient

E:

deformation gradient tensor

f:

specific Helmholtz free energy

F:

force or thermodynamic force

g:

Gibbs free energy per unit volume

G*(ω):

complex stress-strain coefficient

ΔG M :

Flory-Huggins mixing Gibbs free energy

ΔG s :

non-equilibrium contribution to the Gibbs free energy

H:

elastic constant

J*(ω):

complex compliance

J:

steady-state compliance

J1, J:

diffusion fluxes

J s :

entropy flux

k′:

entropic elastic constant

kij :

kinetic constant for the breaking of a macromolecule P j to give P i

Kij :

chemical equilibrium constant defined by the rate k ij /κij

l:

length of the duct

m:

ratio of the partial molar volume of the polymer to one of the solvent

M:

molecular weight

Mn :

number average for the molecular weight of the polymer

Mw :

weight average for the molecular weight of the polymer

M o :

molecular weight of a monomer

n:

number of molecules per unit volume (or number density)

ni :

number of moles of the species i

N:

number of segments in a polymer

NA :

Avogadro's number

ñ 1 :

number of moles of the solvent

ñ 2 :

number of moles of the solute

\(N_i^{(\dot \gamma )}\) :

number of chains with i monomers under a shear

p:

pressure

Δp :

pressure difference

P v :

viscous pressure tensor

P v ij :

components of the viscous pressure tensor

Pi :

chain with i monomers

q:

parameter for defining what kind of averaged molecular weight is used

Q:

flow rate

Qi :

normal modes of the chain

r:

distance to the axis of a duct

R:

constant of gases

R:

the end-to-end vector or the radius of the duct

R i :

the vector from bead i to bead i + 1

s:

specific entropy

T:

absolute temperature

Tc :

critical temperature

Tr:

trace of a tensor

u:

specific internal energy

u:

the local deformation vector

v:

specific volume

v:

velocity vector

V:

symmetric part of velocity gradient

xi :

molar fraction of the species i

W:

configuration tensor

z:

coordination number

α:

parameter of the most probable distribution or the friction coefficient

\(\dot \varepsilon\) :

the extensional rate

εij :

component of the strain tensor

ζ:

friction coefficient

Γ:

gamma function

\(\dot \gamma\) :

the shear rate

γ 0 :

amplitude of the oscillatory shear strain

\(\dot \gamma ^0\) :

amplitude of the oscillatory shear rate

γ′:

infinitesimal displacement gradient in the polymer phase

\(\dot \gamma _c\) :

critical shear rate

\(\dot \gamma _w\) :

shear rate at the wall

γ (0)αβ :

interfacial tension in the absence of flow

\(\dot \gamma _{0.8}\) :

shear rate for which η (\(\dot \gamma\)) is equal to 80% of η0

η:

shear viscosity

ηs :

viscosity of the pure solvent

η 0 :

viscosity for zero shear rate

η*(ω):

complex viscosity

[η]:

intrinsic viscosity

Θ:

theta temperature

ϑ:

the effective flexibility parameter

κ ij :

kinetic constant for the recombination of chains

λ:

correction to the chemical equilibrium reaction by effect of shear

λi :

eigenvalues of the matrix B

μi :

chemical potential of species i

μ 0i :

reference chemical potential of species i

μ (s)i :

non-equilibrium contribution to the chemical potential of i

μ 1s :

non-equilibrium contribution to the solvent chemical potential

μ p2 :

chemical potential under constant P v12

\(\mu _{\dot \gamma 2}\) :

chemical potential under constant \(\dot \gamma\)

μ w2 :

chemical potential under constant W

vi :

stoichiometric coefficient of i

π:

osmotic pressure

πφ :

equilibrium contribution to the osmotic pressure

πel :

non-equilibrium contribution to the osmotic pressure

σ:

the stress tensor

τ:

relaxation time

τd :

disengagement time

φ:

volume fraction of polymer

χ:

Flory's interaction parameter

ψ:

the configurational distribution function

ψ 0 :

the equilibrium distribution function

\(\Psi _1 (\dot \gamma )\) :

the first normal stress coefficient

\(\Psi _2 (\dot \gamma )\) :

the second normal stress coefficient

ω:

angular frequency

Ω:

solid angle

Ω:

orthogonal matrix to diagonalize B

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Authors and Affiliations

  1. Department de Física (Física Estadística), Universitat Autònoma de Barcelona, 08193, Bellaterra, Catalonia, Spain

    D. Jou, J. Casas-Vázquez & M. Criado-Sancho

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  1. D. Jou
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  3. M. Criado-Sancho
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Jou, D., Casas-Vázquez, J., Criado-Sancho, M. (1995). Thermodynamics of polymer solutions under flow: Phase separation and polymer degradation. In: Physical Properties of Polymers. Advances in Polymer Science, vol 120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58704-7_4

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