Amorphous Polymers.pdf

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AMORPHOUS POLYMERS 63

AMORPHOUS POLYMERS

Introduction

Amorphous materials are characterized by the absence of a regular three-

dimensional arrangement of molecules; ie, there is no long-range order. However,

a certain regularity of the structure exists on a local scale denoted as short-range

order. For low molecular weight amorphous materials the structure is character-

ized with respect to the short-range order of the centers of the molecules as well as

the orientational order of the molecular axes. For the case of long-chain amorphous

materials it is necessary to specify an additional structural parameter, namely the

conformation (1) of the chain which depends predominantly on the intramolecular

interactions along the chain (see C ONFORMATIONS AND C ONFIGURATION ).

The structure however is not static but changes continuously as a result

of thermally driven orientational and translational molecular motions. The time

scale of these motions may consist of a few nanoseconds up to several hundred

years. The structure of the amorphous state as well as its time-dependent ﬂuctua-

tions can be analyzed by various scattering techniques such as x-ray, neutron, and

light scattering. The static properties (structure) are probed by coherent elastic

scattering methods, whereas the time-dependent ﬂuctuations are investigated by

inelastic and quasi-elastic neutron scattering, and dynamic light scattering.

Scattering Methods for the Study of Static and Dynamic Properties

X-ray experiments use radiation with a wavelength in the range 10 − 1 – 1 nm (see

X- RAY S CATTERING ). The energy of x-rays is so high that all electrons are excited.

The electric ﬁeld of the incoming wave induces dipole oscillations in the atoms.

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The accelerated charges generate secondary waves that add up at large distances

to the overall scattering amplitude. All secondary waves have the same frequency,

but they may have different phases caused by the different path lengths. Because

of the high frequency it is only possible to detect the scattering intensity, the square

of the scattering amplitude, and its dependence on the scattering angle. Neutron

scattering (qv) experiments allowed for measurements of polymer conformations

at large scales, which were not feasible with x-rays. Neutrons interact with the

nuclei of the atoms whereas x-rays interact with the electrons. The interaction

with matter is different, but the problem of interfering secondary waves is the

same. Instead of the electron density the scattering length density is dealt with.

The essential fact in neutron scattering is the pronounced difference in the scat-

tering amplitude between hydrogen and deuterium, which is important for the

variation of the contrast between the particles and the matrix. Quasi-elastic neu-

tron and light scattering experiments measure the correlation function of the con-

formational ﬂuctuations of the macromolecules at a given length scale. Neutron

scattering monitors the segmental mobility of a polymer chain in the nanometer

and nanosecond region, whereas light scattering reﬂects translational diffusion

of the whole polymer coil.

This article describes the present state of knowledge regarding the struc-

ture of amorphous polymers as obtained from scattering techniques and the corre-

sponding dynamic properties from a structural point of view. A detailed knowledge

of the structure is very important because the thermal, mechanical, viscoelastic,

optical, and even electrical properties are strongly governed by the structure and

its temporal ﬂuctuations.

In order to describe the static structure of the amorphous state as well as

its temporal ﬂuctuations, correlation functions are introduced, which specify the

manner in which atoms are distributed or the manner in which ﬂuctuations in

physical properties are correlated. The correlation functions are related to vari-

ous macroscopic mechanical and thermodynamic properties. The pair correlation

function g ( r ) contains information on the thermal density ﬂuctuations, which in

turn are governed by the isothermal compressibility

κ T ( T ) and the absolute tem-

perature for an amorphous system in thermodynamic equilibrium. Thus the cor-

relation function g ( r ) relates to the static properties of the density ﬂuctuations.

The ﬂuctuations can be separated into an isobaric and an adiabatic component,

with respect to a thermodynamic as well as a dynamic point of view. The adiabatic

part is due to propagating ﬂuctuations (hypersonic sound waves) and the isobaric

part consists of nonpropagating ﬂuctuations (entropy ﬂuctuations). By using in-

elastic light scattering it is possible to separate the total ﬂuctuations into these

components.

Knowledge of the density and orientational correlation functions is not sufﬁ-

cient to characterize the structure completely, since different structures can give

rise to identical correlation functions. Therefore it is necessary to assume mod-

els. A complete description of the structure requires that the spatial arrangement

of the chain elements, the chain conformation, be known. Usually, average val-

ues of the conformation (2) such as the mean square radius of gyration or the

mean square end-to-end distance are determined. Direct structure measurements

involve the interaction between electromagnetic radiation and the substance in

question. A full description of the system will require information about both its

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AMORPHOUS POLYMERS 65

) with properties of the system

is a Fourier or Laplace transformation. The system in real space is characterized by

correlations between various physical properties (eg, particle densities, distances

between atoms, orientations, ﬂuctuations in the local dielectric tensor, pressure

and entropy ﬂuctuations) in terms of the corresponding correlation functions.

The theoretical analysis of the scattered spectrum was ﬁrst presented by

Komarov and Fisher (3) and Pecora (4) independently in 1963. In 1964 the spec-

trum of laser light scattered by dilute solutions of polystyrene latex spheres was

observed (5) and was found to exhibit a lineshape in good agreement with the-

ory. In a typical scattering experiment (6) a monochromatic beam of radiation is

incident on the material; the wave vector is denoted as k 0 and the frequency by

ω 0 . The scattered radiation is recorded as a function of the scattering angle and

frequency shift

, where

k 1 −

k 0 with

sin(

θ/

(1)

k 1 is the wave vector of the scattered radiation,

ω 1 the frequency, and

the wave-

length.

532 nm

Scattering

volume

Nd : YAG

Polarizer

Analyzer

Detector

The distribution of the scattered radiation as a function of q contains infor-

mation about the distribution of atoms or molecules on a molecular level, provided

that the wavelength of the radiation used is of the order of magnitude of the in-

teratomic spacings. This is the case for neutron and x-ray scattering, whereas in

the case of light scattering only integral properties of the structure can be ana-

lyzed because of the large wavelength involved. It is worth mentioning that one

characteristic property of elastic scattering is its coherence. Thus spatial informa-

tion is contained in the phases, and the scattered intensity is determined by the

interference of the scattered waves in front of the detector. Consequently, if q is a

characteristic correlation length in a polymeric system, the obtained information

is an ensemble average over distances of the order of q − 1 . A polarized light scat-

tering experiment with wavelengths 2–3 orders of magnitude longer than neutron

wavelengths, observes averages over much longer correlation lengths. By correctly

chosen q -range, local or global features of the polymers can be studied (7).

static and dynamic properties. Structure information about the time-averaged or

static state is obtained from elastic scattering experiments; ie, the scattered in-

tensity is integrated over all frequencies. Time-dependent or dynamic structures,

on the other hand, can be studied with inelastic scattering techniques; ie, the scat-

tered intensity is analyzed with respect to the frequency. The mathematical tool

which is used to relate the scattering function S ( q ,

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Fig. 1. Depolarized Rayleigh spectra of 1,2-polybutadiene in Aroclor at 353 K. The depo-

larized spectra were ﬁt to either one or a sum of two Lorentzian functions plus a baseline,

considering the overlap of neighboring orders. The integrated intensity is proportional to

the effective optical anisotropy

The method of light scattering with different techniques (8) can cover a

wide dynamic range from 10 5 up to 10 − 12 s. With Fabry–Perot interferometry

the Rayleigh–Brillouin (9) and depolarized spectra (10,11) can be frequency an-

alyzed to reveal the effects of segmental (12) and orientational (13) ﬂuctuations

between 10 − 8 and 10 − 12 s. At longer time scales the scattered light can be an-

alyzed by the photon correlation spectroscopy (pcs) technique. The polarization

of the scattered light and the scattering vector q , which determines the wave-

length of the observed ﬂuctuations, selects and characterizes the observed motion.

Figure 1 shows a typical depolarized Rayleigh spectrum (14) of 1,2-polybutadiene

in Aroclor at 353 K. The depolarized spectra were ﬁt to either one or a sum of

two Lorentzian functions plus a baseline, considering the overlap of neighboring

orders. The integrated intensity is proportional to the effective optical anisotropy

γ eff 2 and is given by following the expression:

I VH =

Af ( n )

ρ ∗ γ

eff

(2)

ρ ∗ is the number

density of the solute. Figure 2 shows depolarized intensity correlation functions

for a poly(styrene- b -1,4-isoprene) block copolymer (15) in the disordered state. The

total molecular weight M n of the copolymer is 3930 and the molecular weight of the

polystyrene (PS) block is 2830. The primary contribution to the spectra comes from

the local segmental motion of the PS in the copolymer and the dispersion broad-

ens with decreasing temperature, which is evident by inspection of the curves in

Figure 2.

γ eff 2 .

where A is a constant, f ( n ) is the product of the local ﬁeld correction and the

geometrical factor 1/n 2 , with n being the refractive index and

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AMORPHOUS POLYMERS 67

parameters used in

the ﬁts are given in the inset. The total molecular weight of the copolymer is M n =

3930

and the molecular weight of the PS block is equal to 2830. The primary contribution to

the spectra comes from the local segmental motion of the PS in the copolymer and the

dispersion broadens with decreasing temperature.

◦ β = 0.38 at 323 K; β = 0.30 at

303 K;

β =

0.25 at 287 K.

The structure of the amorphous state is subjected to time-dependent vari-

ations, because of the existence of translational and/or orientational motions of

both the individual segments and the chain as a whole (16). These motions couple

to the light since they induce variations of the local dielectric constant

( r , t )or

( r , t ). Inelastic and quasi-elastic scattering measurements

allow the determination of the time laws according to which these motions occur

(17). The particular motion detected by the light scattering technique depends on

the polarization of the scattered light and on the scattering vector q , which gives

the wavelength of the observed ﬂuctuations. Light scattering depends on the di-

rections of the polarizations of the scattered radiation and the incident radiation.

Provided that the incident and scattered light are polarized with the electric vec-

tor at right angles to the plane containing k 1 and k 0 , VV scattering is observed.

Frequency shifts are detectable in the case of light scattering, and thus, temporal

ﬂuctuations of the structure can be determined. Individual photons may be scat-

tered with a slight frequency shift. The magnitude and sign of the frequency shift

depend on the velocity and the direction with respect to the incident radiation in

which an individual particle is moving. This frequency shift is referred to as the

Doppler effect . The free motion of the particles (macromolecules) in solution is a

thermal diffusion with no net transport and therefore no net exchange in energy

between the system and the incident light. The scattered light thus consists of a

narrow Lorentzian spectrum of frequencies symmetrically broadened because of

the random Brownian motion of the particles, and centered at the incident fre-

quency of the laser. The time-correlation function and the frequency-dependent

power spectrum are related by Fourier transform. This is usually referred to as

quasi-elastic light scattering (qels), a technique allowing one to extract dynamical

information about the system, ie, the diffusion coefﬁcient. Because of the large

Fig. 2. Depolarized intensity correlation functions for the poly(styrene- b -1,4-isoprene)

block copolymer. Solid curves are the KWW ﬁts to the data and the

the local polarizability

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