Light Scattering from Multicomponent Polymer Systems in Shear Fields: Real-time, In Situ Studies of Dissipative Structures in Open Nonequilibrium Systems

1 Introduction

1.1 General Background

This chapter discusses self-assembly of polymer solutions and solutions of polymer mixtures under external fields, specifically under shear fields, as observed by real-time and in situ light scattering (LS). Although shear-induced structures of solutions of polymer mixtures will be discussed in some parts of this chapter, the discussion can be in principle extended to shear-induced structures of binary polymer mixtures in bulk. This is because the solvent used is a neutral solvent which effectively weakens segmental interactions of polymers as will be detailed later (Section 5.1 ).

Our research theme to be treated here is considered to belong to the general theme on pattern formation in nature. This is because the pattern formation in various systems in nature occurs under the influence of external fields and is inherently related to the self-assembly in the so-called open nonequilibrium systems, i.e., systems that are open to various external...

Glossary of Symbols

$\alpha _{\rm{a}}$

Parameter characterizing for dynamical asymmetry defined by eq. 6

c/c*

Reduced polymer concentration for polymer solutions

c*

Overlap polymer concentration for polymer solutions

C _g

A constant related to the gradient free energy arising from the nonlocality of interactions

CHC

Cahn–Hilliard–Cook

DP

Degree of polymerization

DOP

Dioctylphthalate

$\delta \phi , \delta \phi ({{\bf r}},t)$

Local concentration fluctuations of polymer for polymer solutions or one component of polymer in binary polymer mixtures at position r and time t

D _app

Collective diffusivity for binary mixtures

D _K (K = A or B)

Self-diffusivity of K-th component.

${\rm \Delta} {T}(0)$

Quench depth at $\dot \gamma = 0,{\rm{ }}{\rm \Delta} {T}(0){ = T}_{\rm c} - {T}$ with T _c being critical temperature at $\dot \gamma$= 0

$\eta$

Shear viscosity

$\eta _0$

Zero-shear viscosity

FFT

Fast Fourier transform

GL

Ginzburg–Landau

$\gamma _{{\mathop{\rm int}} }$

Interfacial tension

$\gamma$

Shear strain

$\gamma _0$

Amplitude of oscillatory shear strain

$\dot \gamma$

Shear rate

$\dot \gamma _{{\rm{c,\ single}}}$

Critical shear rate above which shear-induced single-phase formation occurs for binary mixtures

$\dot \gamma _{{\rm{c}},\ x}$

Critical shear rate for shear-induced concentration fluctuations above which scattered intensity along the flow direction (the x axis) starts to increase

$\dot \gamma _{{\rm{c}}z}$

Critical shear rate for shear-induced demixing above which scattered intensity along the neutral axis (the z axis) starts to increase

$\dot \gamma _{\rm{a}}$

Critical shear rate for shear-induced demixing systems above which anomalies are observed in both scattering and rheological properties

${\bf \it\Gamma} _{{\rm{conc}}}$

Relaxation rate for concentration fluctuations in binary mixtures

${\bf \it\Gamma} _{\rm{d}}$

Relaxation rate for deformation of polymer chains

${\bf \it\Gamma} _{{\rm{dis}}}$

Relaxation rate for disentanglement in entangled polymer systems

$\xi$

Mesh size of entangled polymer networks

$\xi _{\rm T}$

Thermal correlation length

$\xi _0$

Average value of $\xi$

$\xi _{{\rm{ve}}}$

Viscoelastic length defined in eq. 13

$(\xi _ \bot )_{\rm{d}}$

Correlation length of string-like domain structures perpendicular to the string axis

$(\xi _ \bot )_{{\rm{fl}}}$

Correlation length for thermal concentration fluctuation along the neutral axis (the z axis)

I(q, t)

Scattering intensity distribution with q at time t after onset of phase separation

k _B

Boltzmann constant

LAOS

Large amplitude oscillatory shear flow (strain)

LS

Light scattering

LSCM

Laser scanning confocal microscopy

$\Lambda (q)$

q-Dependent Onsager kinetic coefficient

$\Lambda (0)$

$\Lambda (q)$ at $q \to 0$

$\Lambda _{\rm{m}} (0)$

Characteristic length of systems undergoing spinodal decomposition in the early stage spinodal decomposition which characterizes Fourier modes of fluctuations having a maximum growth rate

M _W, M _N

Weight and number average molecular weight, respectively

N _K (K = A or B)

Degree of polymerization of K-th component

N ₁

The first normal stress difference

OM

(transmission) Optical microscope

OZ

Ornstein–Zernicke equation (eqs. 28 and 29)

$\omega$

Angular frequency in oscillatory shear strain

PB

Polybutadiene

PI

Polyisoprene

PS

Polystyrene

$\phi ({\rm{\bf r}},t)$

Local concentration of polymer in polymer solutions or one polymer component in binary polymer mixtures at position r and time t

$\phi _K ({\rm{\bf r}},t)$ (K = A or B)

local composition of K-th component in binary mixtures

$\phi _0$

Average of $\phi ({\rm{\bf r}},t)$ or average of $\phi _K ({\rm{\bf r}},t)$ for one of the components (K = A or B) in binary mixtures

$\phi _{\rm{s}}$

Strain phase in oscillatory shear strain

q

Scattering vector or wave vector for Fourier modes

q

Magnitude of q, q = $(4{\rm{\pi }}/\lambda )\sin (\theta /2)$ ($\lambda$ and $\theta$ being wavelength of incident beam and scattering angle in a scattering medium)

q _x ,q _y , q _z

Component of q along flow direction, velocity gradient direction, and neutral direction, respectively

q _m

q at a scattering maximum

q _m(0)

Characteristic wave number which characterizes the Fourier modes of composition (or concentration) fluctuations having a maximum growth rate in early stage of spinodal decomposition

r

Length scale of observation

R _g

Radius of gyration of polymer

SALS

Small-angle light scattering

SBR

Random copolymer of styrene and butadiene, poly(styrene-r-butadiene)

SD

Spinodal decomposition

SQL

Squared Lorentzian function (eqs. 25 and 26) predicted by Debye–Bueche theory [68]

shear-SALS

SALS under shear flow

shear-OM

Transmission optical microscopy under shear flow

$\psi _1$

Coefficient of the first normal stress difference

$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \sigma }$

Shear stress tensor

$\sigma$

Shear stress

$\theta$

Scattering angle

t _c(${\rm \Delta} {T}$)

Characteristic time of mixtures at quench depth ${\rm \Delta} {T}(0)$

$\tau$

Characteristic time in binary mixtures as defined by $\tau = [q_{\rm{m}} ^2 (0)/D_{{\rm{app}}} ]^{ - 1}$

$\tau _p$

$\tau$ for polymer mixtures

$\tau _s$

$\tau$ for small-molecule mixtures

T

Absolute temperature

T _c(0)

Critical temperature of mixtures at $\dot \gamma = 0$

T _c($\dot \gamma$)

Critical temperature of mixtures under shear flow at $\dot \gamma \ne 0$

T _cl

Cloud-point temperature

t _I

Characteristic interface thickness

TDGL

Time-dependent Ginzburg–Landau equation (theory)