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Part III

Liquids

13 Viscoelasticity and Microscopic Motion

in Dense Polymer Systems

Dieter Richter

13.1 Introduction

The mechanical properties of condensed matter are generally divided into

two classes: solids, such as metals or glasses, behave elastically; for small

deformations the stress is proportional to the strain – Hooke’s law is valid.

Liquids display viscous behavior; in this case the stress is proportional to the

change in strain, Newton’s law persists. In contrast, the mechanical dynam-

ical properties of polymers are extraordinarily versatile. Depending on the

temperature or load time, the same polymer may display vitreous, rubber-

like or ﬂuid behaviour. This wide range of behaviour is known as viscoelastic.

In particular, the dynamic modulus of long-chain linear polymers dis-

plays a plateau regime which expands with growing chain length [1]. In this

plateau region, the stress is proportional to strain – although a liquid, the

polymer melt behaves elastically. For long times the material yields and vis-

cous behavior takes over. This behaviour reminds us of the rubber elasticity

of chemically cross-linked polymers. The elastic modulus of rubbers is pro-

portional to the temperature and inversely proportional to the mesh size of

the network. The proportionality to temperature is caused by entropic forces

resulting from the conformational entropy of the coiled chains between the

network points. If chain pieces are stretched, an entropic force originating

from the reduction of free energy acts with the aim of restoring the more

probable coiled state for these stretched segments.

Based on this analogy to rubber elasticity it is natural to assume that the

elastic behaviour of polymer melts is caused by network-like entanglements of

the polymer chains, so that the melt becomes a temporary network. The role

of the network points is taken over by entanglements. The plateau modulus

is ascribed to the rubber elastic modulus of this temporary network. On

this assumption it is possible to estimate the mean distance between the

hypothetical entanglement points in the melt from the value of the plateau

modulus. The resultant entanglement distances d are typically of the order

of d =40− 100

A. Compared with the two characteristic length scales of

a polymer, the monomer length  ∼ 5

A and the coil end-to-end distance

∼ 1000

A, the entanglement distance deﬁnes an intermediate length scale

of a dynamic nature.

514 Dieter Richter

If a solid is subjected to a mechanical strain, then the individual atoms

or molecules move in their local potential and take on a new equilibrium

position. A macroscopic deformation at a polymer, in contrast, causes com-

plex molecular rearrangements, starting from individual bonds up to and

including the molecular dimensions. On the scale of the individual bond,

these rearrangements proceed at a speed similar to that in a normal solid.

This local bond dynamics is governed by the local potentials and occurs on

length scales of a few

Angstrøms and times from several picoseconds upwards

and is of particular interest in connection with glass transition phenomena.

For intermediate times corresponding to spatial distances shorter than the

entanglement distance we expect motional mechanisms which are driven by

entropic forces. These motions are described by the so-called Rouse model [2].

As the spatial extension of motion increases and reaches dimensions of the en-

tanglement distance, the freedom of motion of the chain is greatly restricted.

The temporary network leads to a localization of the chain, which dominantly

moves like a snake along its own proﬁle due to the network, which restricts

lateral motions. For very long times, the chain leaves its original topolog-

ical restrictions behind – the plateau modulus relaxes [3]. This concept is

essentially contained in the reptation model of de Gennes. All associated

movements, starting from lengths of about 1

A and times in the order of

magnitude of 10

−11

s up to the size of a molecule of typically 1000

A, deter-

mine the viscoelastic properties of a polymer system.

Applying quasielastic neutron scattering (QENS) the aim of the stud-

ies presented here was a space-time analysis of the above-described mo-

tions [4–23], in order to achieve molecular understanding of the mechanical

properties of polymer melts. After a short introduction to the neutron spin-

echo technique, which complements the presentation of quasielastic scattering

given in Chaps. 2 and 3, ﬁve topics will be dealt with. First of all, experi-

ments on the local bond dynamics and segmental motion giving rise to the α-

and β-relaxation in polymer systems will be reported. Thereafter, the entropi-

cally driven Rouse motion will be discussed. Subsequently, on long chains this

chapter presents direct experimental observation of the reptation process. At

the end we address the transition regions: First we report data from the cross

over from universal entropic Rouse dynamics to local dynamics. Then we ex-

amine the entanglement transition and display how topological restrictions

manifest themselves if the polymer chain length increases.

13.2 The Neutron Scattering Method

Scattering methods play a prominent role if spatial structures are to be ex-

plored on the atomic and molecular scale. The two most important probes

are photons in the X-ray region and neutrons. While photons with appropri-

ate wavelength possess energies of the order of keV, neutrons with de Broglie

13 Viscoelasticity and Microscopic Motion in Dense Polymer Systems 515

Fig. 13.1. Schematic representation of a neutron scattering experiment; Mono:

monochromator; Anal: analyzer; Det: detector; E

, E

, incident and scattered neu-

tron energies; k

, k

: corresponding wave vectors; ∆E: energy transfer; Q:momen-

tum transfer; 2ϑ: scattering angle.

wavelengths in the

A region have thermal or subthermal energies. The neu-

tron probe thus permits a simultaneous space and time analysis of the thermal

motion in condensed matter [24] (see also Chaps. 2 and 3). Whereas the X-

ray photon provides information about the molecule location, the neutron

reveals the location of an atom and its direction, type and rate of motion.

Unlike photons, neutrons are scattered on the atomic nucleus. This provides

the possibility of selective atomic labelling by isotope exchange. Especially

the scattering properties of protons and deuterons, described by their scat-

tering lengths (b

= −0.34 · 10

−12

m, b

=0.66 · 10

−12

m), are particularly

diﬀerent. As a unique feature therefore neutrons provide the opportunity to

study synthetic and biological macromolecules among their equals if labelled

by deuteration. For example, the coil conformation of the polymer in the melt

and in the amorphous state was only conﬁrmed experimentally in this way.

Figure 13.1 shows the schematic setup of a neutron scattering experiment.

A monochromator is used to select neutrons with a deﬁned energy, E

,and

a wave vector, k

, from the neutron spectrum of a neutron source. These

neutrons are scattered on a specimen and analysed under the scattering angle

2ϑ. The analyser chooses neutrons with a wave vector k

andanend-point

energy E

. These neutrons are identiﬁed in a detector. Momentum transfer

Q = (k

− k

) and energy transfer ∆E = E

− E

correspond to distances

∼

=2π/Q and to times t

∼

= h/∆E.

A neutron scattering experiment generally measures the space-time Fourier

transform of the pair-correlation function S(Q, ω) or, in speciﬁc cases, only

the spatial Fourier transform S(Q, t)(ω = ∆E) (cf. Sect. 3.2 in Chap. 3). In

classical approximation, the pair-correlation function means the conditional

probability of ﬁnding an atom j at location r and time t, if another atom i was

at location r =0attimet =0.Fori = j the self-correlation function is ob-

tained, which measures the time-dependent thermally averaged mean square

516 Dieter Richter

displacement (r

(0) −r

(t))

. For the time-dependent scattering function in

the so-called Gaussian approximation we have

S(Q, t)=



exp



−Q

(r

(0) − r

(t))





. (13.1)

In the case of a labelled polymer, r

and r

denote the spatial monomer

coordinates and N is the number of monomers in a chain. The sum extends

over all monomers in a chain.

13.2.1 The Neutron Spin-Echo Technique

Versus Conventional Scattering

The unique property of the neutron spin-echo (NSE) technique [25,26] is its

ability to directly measure energy changes of the neutron during scattering.

This distinguishes NSE from conventional scattering techniques which, as

explained above, proceed in two steps: ﬁrst monochromatization of the inci-

dent beam and then analysis of the scattered beam. Energy and momentum

changes during scattering are determined by taking diﬀerences between two

measurements. In order to achieve high energy resolutions with these conven-

tional techniques, a very narrow energy interval must be selected from the

relatively low-intensity neutron spectrum of the source. Conventional high-

resolution techniques therefore always have diﬃculties with low intensity.

Unlike these conventional techniques, NSE measures the neutron veloci-

ties of the incident and scattered neutrons using the Larmor precession of the

neutron spin in an external magnetic ﬁeld. The neutron spin vector acts like

the hand of an internal clock, which is linked to each neutron and stores the

result of the velocity measurement at the neutron itself. The velocity mea-

surement is thus performed individually for each neutron. For this reason, the

velocities prior to and after scattering on one and the same neutron can be

directly compared and a measurement of the velocity diﬀerence during scat-

tering becomes possible. The energy resolution is thus decoupled from the

monochromatization of the incident beam. Energy resolutions of the order of

−5

may be achieved with an incident neutron spectrum of 20% bandwidth.

13.2.2 Neutron Spin Manipulations with Magnetic Fields

The motion of neutron polarization – it is the quantum mechanical expecta-

tion value of neutron spin – is described by the Bloch equation

P =

|g|µ



[B × P ] (13.2)

where g is the g-factor of the neutron (g = −3.82), µ

the nuclear magne-

ton and B the ﬂux density of the magnetic ﬁeld H.Thefactorgµ

/ is

13 Viscoelasticity and Microscopic Motion in Dense Polymer Systems 517

 = 180°

Fig. 13.2. Spin angular operations in neutron spin-echo technique; a) arrangement

of a Mezei coil for a rotation of the neutron spin by the angle π; b) motion of neutron

polarization during π angular operation; c) motion of neutron polarization during

a π/2 rotation, after [25].

equal to the magnetogyric ratio γ commonly used in NMR (see Chaps. 9,

10 and 17; for the neutron γ = − 1.83 · 10

−1

). Equation (13.2) is the

basis for manipulation of the neutron polarization by external ﬁelds. Let us

discuss two simple spin angular operations. We consider a neutron beam,

which propagates with a polarization parallel to the direction of propagation

in the z-direction. A magnetic guide ﬁeld parallel to z stabilizes the polar-

ization. First of all, we will explain the so-called π-coil, which reverses two

components of the neutron spin. Its principle is shown in Figs. 13.2a and b.

A ﬂat, long, rectangular coil, a so-called Mezei coil, is slightly tilted with

respect to the x,y plane. A magnetic ﬂux B

is generated so that the resultant

ﬂux B

= B

+ B

points to the x-direction. A neutron spin entering this

ﬁeld begins to rotate about the x-axis. During a time t = d/v, d being the

coil thickness and v the neutron velocity, a phase angle φ = ω

t is passed.

With the Larmor frequency ω

= g(µ

/)B

and v = h/(λm) we ﬁnd

φ =



2π|g|µ



λB

d (13.3)

where m is the neutron mass and λ the neutron wavelength. φ is thus given by

the line integral

Bds and is proportional to the wavelength. If we take, for

example, a coil thickness of d = 1 cm and a neutron wavelength of λ =8

518 Dieter Richter

aﬁeldB

of 0.85 mT is needed to rotate the neutron spin by 180

◦

. Such a

spin rotation operation is shown schematically in Fig. 13.2b. Thereby, the

components of polarization x,y,z are transferred to x,−y,−z.

The second important spin angular operation is the 90

◦

rotation where

the polarization is transformed from the z-tothex-direction or vice versa.

A Mezei coil in the x,y-plane is adjusted such that the resultant ﬁeld points

exactly to the direction of the bisectrix of the angle between x and z.A

180

◦

rotation about this axis transforms the z-component of polarization to

the x-direction. At the same time, the sign of the y-component is inverted

(Fig. 13.2c).

13.2.3 The Spin-Echo Principle

The basic experimental setup of a neutron spin-echo spectrometer is shown

in Fig. 13.3. A velocity selector in the primary neutron beam selects a wave-

length interval of about 10%–20% width. The spectrometer oﬀers a primary

and a secondary neutron ﬂight path passing through the precession ﬁelds B

and B



. Prior to beginning the ﬁrst ﬂight path, the neutron beam is polarized

in the forward direction with the aid of a supermirror polarizer. A ﬁrst π/2

coil rotates the polarization to the x-direction perpendicular to the direction

of propagation. Beginning with this well-deﬁned initial condition, the neu-

trons precess in the precession ﬁelds. Without the eﬀect of the π-coil each

neutron would go through a phase angle φ ∝

Bds (see (13.3)). Since the

wavelengths of the neutrons are distributed over a wide range, the phase an-

gle for each neutron would be diﬀerent in front of the second π/2coilandthe

beam would be depolarized on a disk perpendicular to z.Theπ-coil, which

is exactly positioned at the value of half the ﬁeld integral, prevents this ef-

fect. Let the neutron spin pass through the phase angle Ω

=2πn + ∆φ

on its way to the π-coil. The eﬀect of the π-coil transforms the angle ∆φ

to −∆φ

. In a symmetric arrangement (both ﬁeld integrals in front of and

behind the π-coil are identical) the neutron spin passes through a second

phase angle φ

= φ

=2nπ + ∆φ

. The spin transformation at the π-coil

thus just compensates the angles ∆φ

, and in front of the second π/2coil

the neutron spin points again to the x-direction irrespective of the velocity.

This eﬀect is called spin focusing or spin echo and is also the basis of various

NMR techniques to study diﬀusion (see Chap. 9, Sect. 9.3; Chap 10, Sect.

10.3; Chap. 17, Sect 17.2).

The second π/2 coil projects the x-component of polarization onto the

z-direction. It is then determined by a subsequent supermirror analyser and

detector. In a spin-echo experiment, the specimen is positioned near the π-

coil. With the exception of losses due to ﬁeld inhomogeneities, the polariza-

tion is maintained in the case of elastic scattering. If, however, the neutron

energy is changed due to inelastic scattering processes in the specimen, then

the neutron wavelength is modiﬁed from λ to λ



= λ + δλ.Inthiscase,

the phase angles φ

and φ

are not compensated anymore. The second π/2