Determining the Glass Transition in Polymer Melts ......transition in general.2,3 Instead of...

CHAPTER 1

Determining the Glass Transitionin Polymer Melts

Wolfgang Paul

Institut fur Physik, Johannes Gutenberg-Universitat, Mainz,Germany

INTRODUCTION

In the last 15 years, computer simulation studies of the glass transition inpolymer melts have contributed significantly to advance our understanding ofthis phenomenon, which is at the same time of fundamental scientific interestand of great technical importance for polymer materials, most of which areamorphous or at best semi-crystalline. This progress has been possible, onthe one hand, because of improved models and simulation algorithms and,on the other hand, because of theoretical advances in the description of thestructural glass transition in general.1 Much of this development has been mir-rored in a series of conferences on relaxations in complex systems the proceed-ings of which might serve as a good entry point into the literature on the glasstransition in general.2,3

Instead of providing a detailed overview of all simulation work per-formed on the glass transition in polymer melts, this review has two goals.The first goal is to provide a novice to the field with the necessary backgroundto understand the model building and choice of simulation technique for stu-dies of the polymer glass transition. In particular, a novice modeler needs to beaware of the strengths and limitations of the different approaches used in thesimulation of glass-forming polymers and to be able to judge the validity of theoriginal literature. The second goal is to present a personal view of the

Reviews in Computational Chemistry, Volume 25edited by Kenny B. Lipkowitz and Thomas R. Cundari

Copyright � 2007 Wiley-VCH, John Wiley & Sons, Inc.

1

COPYRIG

HTED M

ATERIAL

contribution that computer simulations have made to our understanding ofdifferent aspects of the polymer glass transition, ranging from thermodynamicto dynamic properties. This part of the review will present our current under-standing of glass transitions in polymeric melts based on simulation, experi-ment, and theory. We will illustrate this understanding based mainly on ourown contributions to the field.

In the next section, a short summary of the phenomenology of the glasstransition is presented. The following section on models then explains the var-ious types of models employed in the simulation of polymer melts, and theensuing section on simulation methods introduces the algorithms used forsuch simulations. We will then describe simulation results on concepts relatingto the thermophysical properties of the polymer glass transition. Finally, themain section of this review will present an overview of simulations of the slow-down of relaxation processes in polymer melts upon approaching the glasstransition, and in the conclusions, we summarize what has been learned abouthow to identify the glass transition in polymer melts.

PHENOMENOLOGY OF THE GLASS TRANSITION

The defining property of a structural glass transition is an increase of thestructural relaxation time by more than 14 orders in magnitude without thedevelopment of any long-range ordered structure.1 Both the static structureand the relaxation behavior of the static structure can be accessed by scatteringexperiments and they can be calculated from simulations. The collective struc-ture factor of a polymer melt, where one sums over all scattering centers M inthe system

SðqÞ ¼ 1

M

XM

i;j¼1

hexp½i~q � ð~ri �~rjÞ�i ½1�

resembles the structure factor of small molecule liquids (we have given here asimplified version of a neutron structure factor: all scattering lengths have beenset to unity).

In Figure 1, we show an example of a melt structure factor taken from amolecular dynamics simulation of a bead-spring model (which will bedescribed later). The figure shows a first peak (going from left to right), theso-called amorphous halo, which is a measure of the mean interparticle dis-tance in the liquid (polymer melt). Upon lowering the temperature to the glasstransition, the amorphous halo shifts to larger momentum transfers as themean interparticle distance is reduced by thermal expansion. The amorphoushalo also increases in height, which indicates smaller fluctuations of the meaninterparticle distance, but no new structural features are introduced by thiscooling.

2 Determining the Glass Transition in Polymer Melts

The thermal expansion, however, changes behavior at the glass transi-tion, which is a phenomenon that was first analyzed in detail in a careful studyby Kovacs.4 In the polymer melt, the thermal expansion coefficient is almostconstant, and it is again so in the glass but with a smaller value. At the glasstransition, there is therefore a break in the dependence of density on tempera-ture that is the foremost thermophysical characteristic of the glass transition.

The decay of the structural correlations measured by the static structurefactor can be studied by dynamic scattering techniques. From the simulations,the decay of structural correlations is determined most directly by calculatingthe coherent intermediate scattering function, which differs from Eq. [1] by atime shift in one of the particle positions as defined in Eq. [2]:

Sðq; tÞ ¼ 1

M

XM

i;j¼1

hexp½i~q � ð~riðtÞ �~rjð0ÞÞ�i ½2�

The Fourier transform of this quantity, the dynamic structure factor Sðq;oÞ, ismeasured directly by experiment. The structural relaxation time, or a-relaxationtime, of a liquid is generally defined as the time required for the intermediatecoherent scattering function at the momentum transfer of the amorphous haloto decay to about 30%; i.e., Sðqah; taÞ ¼ 0:3.

The temperature dependence of the a time scale exhibits a dramatic slow-down of the structural relaxation upon cooling. This temperature dependence

0 5 10 15 20q

0

1

2

3

4

S(q

)

T=0.46T=0.52T=1.0

Figure 1 Melt structure factor for three different temperatures (given in Lennard–Jonesunits) taken from a bead-spring model simulation. In the amorphous state (melt andglass), the only typical length scale is the next neighbor distance giving rise to theamorphous halo (first sharp diffraction peak) around q ¼ 6:9 for this model.

Phenomenology of the Glass Transition 3

qualitatively agrees with that of the melt viscosity. This macroscopic measure ofthe relaxation time in the melt serves to define the so-called viscosimetric glasstransition Tg as the temperature at which the viscosity is 1013 Poise. This resultcorresponds to a structural relaxation time of approximately 100 s. In Figure 2,we show three typical temperature dependencies of the viscosity in the form ofan Angell plot.5 The upper curve is an Arrhenius law defining so-called ‘‘strongglass formers.’’ The two other curves follow Vogel–Fulcher laws (Eq. [3])observed for ‘‘fragile glass formers,’’ a category to which most polymeric sys-tems belong, displaying a diverging viscosity at some temperature T0 < Tg.Around Tg, the relaxation time of fragile glass formers increases sharply. Thedefinition of Tg is thus based on the fact that at this temperature the system fallsout of equilibrium on typical experimental time scales. As a result of this fallingout of equilibrium, one also observes a smeared-out step in the temperaturedependence of the heat capacity close to Tg defining the calorimetric Tg (similarto the behavior of the thermal expansion coefficient). The calorimetric Tg andthe viscosimetric Tg need not agree exactly. For crystallizable polymers, one candefine a ‘‘configurational entropy’’ of the polymer melt by subtracting the entro-py of the corresponding crystal from the entropy of the melt. A monotonousdecrease is predicted in the configurational entropy to a value at Tg, which isabout one third of the corresponding value of the configurational entropy atthe melting temperature of the crystal.6 Extrapolating to lower temperatures,one finds the configurational entropy to vanish at the Kauzmann temperatureTK,7 which is typically 30–50 K lower than Tg.5 It is interesting to note that

0 0.2 0.4 0.6 0.8 1T / T

g

0.01

1

100

10000

1e+06

1e+08

1e+10

1e+12η

ArrheniusVF: T

0= 0.8 T

g

VF: T0= 0.9 T

g

Figure 2 Sketch of typical temperature dependencies of the viscosity Z of glass-formingsystems. The viscosimetric Tg of a material is defined by the viscosity reaching 1013

Poise. Strong glass formers show an Arrhenius temperature dependence, whereas fragileglass formers follow reasonably well a Vogel–Fulcher (VF) law predicting a divergingviscosity at some temperature T0.


TK is often close to the Vogel–Fulcher temperature T0 discussed in connectionwith Figure 2, which is determined by fitting the Vogel–Fulcher relation5–8 tothe temperature dependence of the structural relaxation time of the melt9 usingEq. [3]:

t ¼ t1 exp½Eact=kBðT � T0Þ� ½3�

where t1 is a time characterizing microscopic relaxation processes at hightemperatures and Eact is an effective activation energy.

Up to this point the phenomenological characterization of the glass tran-sition is the same for a polymer melt and for a molecular liquid. In a polymermelt, however, one must also have knowledge of both the conformationalstructure and the relaxation behavior of a single chain to characterize the sys-tem completely, be it in the melt state or in the glassy state. Flexible linearmacromolecules in the melt adopt a random coil-like configuration; i.e., theirsquare radius of gyration is given by10–12 Eq. [4]:

R2g ¼

C1‘2N

6¼ Nb2

6½4�

where N ðN � 1Þ is the degree of polymerization and ‘ is the length of a seg-ment. The characteristic ratio C1 describes short-range orientational correla-tions among subsequent monomer units along the backbone of the polymerchain, and b ¼

ffiffiffiffiffiffiffiffiC1p

‘ is the statistical segment length of the chain. On inter-mediate length scales, the structure of a polymer coil is well described by theDebye function10 of Eq. [5]:

SpðqÞ ¼1

N

XN

i;j¼1

hexp½i~q � ð~ri �~rjÞ�i ¼ NfDðq2R2gÞ

fDðxÞ ¼2

x2½expð�xÞ � 1þ x�

½5�

where qb� 1 is assumed for the momentum transfer and we again set all scat-tering lengths to unity.

In the dense melt, these coils interpenetrate each other. Thus, their diffu-sive motion is slow even at temperatures far above the glass transition. If thechain length N is smaller than the ‘‘entanglement chain length’’ Ne, abovewhich reptation-like behavior sets in,12–15 the relaxation time describinghow long it takes a coil to renew its configuration is given by the Rouse time

tR ¼ �ðTÞN2C1‘2=ð3p2kBTÞ ½6�

where �ðTÞ is the friction coefficient experienced by the segments of the chain intheir Brownian motion, kB is Boltzmann’s constant, and T is the temperature.

Phenomenology of the Glass Transition 5

The Rouse model12 that yields Eq. [6] also shows that the self-diffusion constantof the chains scales inversely with chain length

DN ¼ kBT=ðN�ðTÞÞ ½7�

whereas the melt viscosity is proportional to the chain length

Z ¼ c�ðTÞb2N=36 ½8�

with c being the number of monomers per volume.14,15

Ample experimental evidence exists10–13 that Eqs. [4]–[8] capture theessential features of (nonentangled) polymer chains in a melt; however, recentsimulations and experiments16,17 have shown that the relaxation of coils onlength scales smaller than Rg is only qualitatively described by the Rouse mod-el. The glass transition manifests itself in the temperature dependence of thesegmental friction coefficient �. Within the Rouse model, this quantity cap-tures the influence of the specific chemistry on the dynamics in the melt,whereas the statistical segment length b captures its influence on the staticproperties. This result explains the two types of models used to study the prop-erties of polymer melts (the glass transition being one of them). Coarse-grainedmodels, like a bead-spring model in the continuum or lattice polymer models,can reproduce the chain length scaling of static and dynamic properties inpolymer melts when they correctly capture the determining physics. That phy-sics involves the excluded volume between all segments and the connectivity ofthe chains. Chemically realistic models are needed when one either tries toreproduce experimental data quantitatively or to describe polymer propertieson length and time scales that are still influenced by the detailed chemistry.

A particular characteristic feature of dynamic processes in the vicinity ofthe glass transition is the ubiquity of the Kohlrausch–Williams–Watts (KWW)stretched exponential relaxation:1,7–9

fðtÞ / exp½�ðt=tÞb�; 0 < b < 1 ½9�

Relaxation functions fðtÞ, which are observable via mechanical relaxation,dielectric relaxation, multidimensional nuclear magnetic resonance (NMR)spectroscopy, neutron-spin echo scattering, and so on, can be described intheir long-time behavior by Eq. [9]. The exponent b typically lies in the range0:3 � b < 1 and depends on what is relaxing. Although the relaxation time tdepends strongly on temperature, b is often approximately independent oftemperature in some temperature interval. In this regime, fðtÞ exhibits a scal-ing property called the ‘‘time-temperature superposition principle.’’7–9

Polymers are very good glass formers, with a few notable exceptions.For some polymers, such as atactic polypropylene or random copolymers like


cis-trans polybutadiene, no possible crystalline state is known, so in these casesit is not clear at all whether we can speak about a super-cooled liquid onapproaching the glass transition. Even when there is an ordered ground state(crystalline or only liquid crystalline) for a specific polymer, we can easily under-stand that a kinetic hindrance for ordering exists. In order to crystallize, a chainmust change its random-coil-like state in favor of one of its possible energeticground state conformations. Because of the packing constraints in a densemelt, this must happen (presumably) in a synchronized fashion with the sur-rounding chains. Thus, it is understandable that polymers are hard to crystallize.Accordingly, whether no known ordered state exists or whether that state is onlykinetically inaccessible, it is easy to observe and measure metastable thermalequilibrium properties of polymer melts (e.g., specific heat or entropy) fromthe high-temperature melt to the low-temperature glass.

MODEL BUILDING

Our aim is to better understand the glass transition phenomenon in poly-mer melts by using computer simulations. The discussion of the glass transi-tion phenomenology in the previous section made it clear that we candistinguish several levels of specificity in our computational quest: (1) Wecan try to model generic features of the structural glass transition, i.e., thosefeatures that are independent of whether we are considering a polymer melt or,e.g., an organic liquid. (2) We can try to determine features of the structuralglass transition that are specific to polymeric materials as compared with, e.g.,silica glasses. (3) We can try to understand quantitatively the glass transitionfor a specific polymeric material. If the aim of our work falls into category (1)and partly (2), it is most efficient from the modeling perspective to use coarse-grained models that capture only generic polymeric properties like monomerconnectivity and excluded volume. If the aim of our work falls into category(3) and partly (2), we will need to employ a chemically realistic model forwhich quantitative input on the local geometry and energetics, i.e., a well-cali-brated force field, is required. Below we consider first chemically realistic mod-els and then we describe two classes of coarse-grained models.

CHEMICALLY REALISTIC MODELING

If we are aiming for a quantitatively correct prediction of the behaviorand properties of a specific polymer, we need to employ an optimized andcarefully validated force field for this specific polymer. In the literature oneoften finds simulation work using force fields that do not fulfill these criteriabut where instead the authors use ‘‘polymer-xy-like’’ models. Although thesemodels fail to reproduce the properties of the polymer they claim to model

Chemically Realistic Modeling 7

quantitatively, the qualitative conclusions drawn from these simulations areoften valid, especially when they concern more general polymeric properties.However, even qualitative conclusions pertaining only to a specific polymer orto a class of similar polymers can be problematic when derived from simula-tions employing inaccurate or unvalidated potentials. Various forms of classi-cal potentials (force fields) for polymers can be found in the literature18–23 andhave been reviewed in this book series.24–28 We are concerned in this chapterwith reproducing the static, thermodynamic, and dynamic (transport andrelaxational) properties of non-reactive organic polymers, and for this reason,the potential must represent accurately the molecular geometry, nonbondedinteractions, and conformational energetics of the macromolecules of interest.

The classical force field represents the potential energy of a polymerchain, made of N atoms with coordinates given by the set f~rg, as a sum of non-bonded interactions and contributions from all bond, valence bend, and dihe-dral interactions:

Vðf~rgÞ ¼ Vnbðf~rgÞ þ Vpolðf~rgÞ¼ Vnbðf~rgÞ þ

X

bonds

VbondðrijÞ þX

bends

VbendðyijkÞ þX

dihedrals

Vtorsð�ijklÞ

½10�

More complicated cross-terms between the different intramolecular degrees offreedom are also employed in some force fields, but we will not consider them inthe following. The dihedral term may also include four-center improper torsionor out-of-plane bending interactions that occur at sp2 hybridized centers.29

The nonbonded interactions commonly consist of a sum of two-bodyrepulsion and dispersion energy terms between atoms that are often of theLennard–Jones form in addition to the energy from the interactions betweenfixed partial atomic or ionic charges (Coulomb interaction)

Vnbðf~rgÞ ¼XM

i;j¼1

4Esrij

� �12

� srij

� �6" #

þ qiqj

4pE0rij½11�

The dispersion interactions are weak compared with repulsion, but theyare longer range, which results in an attractive well with a depth E at an inter-atomic separation of smin ¼ 21=6s. The interatomic distance at which the netpotential is zero is often used to define the atomic diameter. In addition to theLennard–Jones form, the exponential-6 form of the dispersion–repulsion interaction,

Vexp�6ðf~rgÞ ¼ 1

2

XM

i;j¼1

Aij expf�Bijrijg �Cij

r6ij

½12�


is often used in atomistic models. Nonbonded interactions are typicallyincluded in the force field calculations for all atoms of different moleculesand for atoms of the same molecule separated by more than two bonds orby three bonds when the nonbonded ‘‘1-4’’ interaction has been included inthe parameterization of an effective torsional interaction.29

The Coulomb interaction is long-range, which necessitates use of specialnumerical methods for efficient simulation.30 When one tries to understand theglass transition in a chemically realistic model, these long-range Coulombinteractions add further numerical overhead so that the most extensive glasstransition simulations of realistic models were done for apolar molecules.

In atomistic force fields, the contributions from bonded interactionsincluded in Eq. [10] are commonly parameterized as

VbondðrijÞ ¼1

2kbond

ij ðrij � r0ijÞ

2 ½13�

VbendðyijkÞ ¼1

2kbend

ijk ðyijk � y0ijkÞ

2 � 1

2k0

bendijk ðcos yijk � cos y0

ijkÞ2 ½14�

Vtorsð�ijklÞ ¼1

2

X

n

ktorsijkl ½1� cosðn�ijklÞ� or ½15�

Vtorsð�ijklÞ ¼1

2

X

n

koopijkl f

2ijk

Here, r0ij is an equilibrium bond length and y0

ijk is an equilibrium valence bendangle, whereas kbond

ij , kbendijk , ktors

ijkl ðnÞ, and koopijkl are the bond, bend, torsion, and

out-of-plane bending force constants, respectively. The indices indicate which(bonded) atoms are involved in the interaction. These geometric parametersand force constants, combined with the nonbonded parameters qi, E, and s,constitute the classical force field for a particular polymer.

Although there are existing standard force fields in the literature likeAMBER,18 OPLS-AA,20 COMPASS,21 CHARMM,22 and PCFF23 to namebut a few (see also Refs. 24–28), one will typically find that they are only aqualitative or at best a semi-quantitative representation of a polymer onemight want to study. The quantitative modeling of a given polymeric materialhas to start from high-level quantum chemistry calculations as the best sourceof molecular level information for force field validation and parameterization.Although such calculations are not yet possible on high polymers, they are fea-sible on small molecules that are representative of polymer repeat units and foroligomers. These calculations can provide the molecular geometries, partialcharges, polarizabilities, and the conformational energy surface needed foraccurate prediction of structural, thermodynamic, and dynamic properties ofpolymers. A general procedure for deriving quantum chemistry–based poten-tials can be found in the literature.29,31,32 The intermolecular dispersion inter-actions can also, to a certain extent, be determined from these calculations.

Chemically Realistic Modeling 9

However, it has turned out that the most accurate way of fixing these para-meters is through matching of simulated phase equilibria to those derivedfrom experiment.33 As a final step, the potential, regardless of its source,should be validated through extensive comparison with available experimentaldata for structural, thermodynamic, and dynamic properties obtained fromsimulations of the material of interest, closely related materials, and modelcompounds used in the parameterization. The importance of potentialfunction validation in simulation of real materials cannot be overemphasized.

For nonpolar, simple hydrocarbon chains that we will discuss later wecan employ a simple force field of the form

Vðf~rgÞ ¼X

i

VðliÞ þX

j

VðyjÞ þX

k

Vð�kÞ þX

n;m

VnbðrnmÞ ½16�

where the sums run over all bonds, bends, torsions, and nonbonded interactingatoms, respectively. One often does not treat the hydrogen atoms explicitly insimulations of hydrocarbon chains but instead combines them with the carbonatoms to which they are bound to create ‘‘united atoms.’’ This approximationnot only reduces the number of force centers for the calculation of the non-bonded interactions, but it also removes the highest frequency oscillations(C–H bond length and H–C–H and H–C–C bond angles) from the model.This approximation works well when one wants to study structure and relaxa-tional properties in amorphous polymers without any specific local interac-tions (i.e., strong electrostatic interactions or hydrogen bonding). In thelatter cases, internal degrees of freedom of the united atoms and a model fortheir interaction may be added, but no reliable way exists so far to determinethe parameters entering such a description quantitatively for a given polymer,so one generally loses the ability to obtain quantitative predictions using suchmodels. A final approximation often employed in large-scale polymer simula-tions is to neglect the C–C bond length oscillations and to perform the simula-tion with constrained bond lengths.34 The approximations discussed in the lastparagraph are motivated by computational expediency. However, they reflectour understanding of the relevant physical processes that must be included inthe computer simulation for us to obtain a quantitative reproduction of thestructure and dynamics of a realistic polymer melt. Such approximations areimposed on us by the huge spread of relaxation times one has to cover in thesimulation, which range from local relaxations to conformational changes ofunentangled chains requiring substantial computational efforts when one isstriving to perform simulations in thermodynamic equilibrium. The simulationstudies of dynamic processes are generally conducted using moleculardynamics (MD) methods. Equilibrating the starting configurations for thesestudies, however, can profit from the use of Monte Carlo (MC) techniqueswhere moves generating global conformational rearrangements are included.35


COARSE-GRAINED MODELS

Coarse-grained polymer models neglect the chemical detail of a specificpolymer chain and include only excluded volume and topology (chain connec-tivity) as the properties determining universal behavior of polymers. They canbe formulated for the continuum (off-lattice) as well as for a lattice. For allcoarse-grained models, the repeat unit or monomer unit represents a sectionof a chemically realistic chain. MD techniques are employed to studydynamics with off-lattice models, whereas MC techniques are used for the lat-tice models and for efficient equilibration of the continuum models.36–42 Atutorial on coarse-grained modeling can be found in this book series.43

Coarse-Grained Models of the Bead-Spring Type

These models retain the form of the nonbonded interaction used in thechemically realistic modeling, i.e., they use either an interaction of theLennard–Jones or of the exponential-6 type. The repulsive parts of thesepotentials generate the necessary local excluded volume, whereas the attractivelong-range parts can be used to model varying solvent quality for dilute orsemi-dilute solutions and to generate a reasonable equation-of-state behaviorfor polymeric melts.

The inclusion of chain connectivity prevents polymer strands from cross-ing one another in the course of a computer simulation. In bead-spring poly-mer models, this typically means that one has to limit the maximal (or typical)extension of a spring connecting the beads that represent the monomers alongthe chain. This process is most often performed using the so-called finitelyextensible, nonlinear elastic (FENE) type potentials44 of Eq. [17]

UFðlÞ ¼ �1

2kl2max ln½1� ðl=lmaxÞ2� 0 � l � lmax ½17�

but also with harmonic spring length potentials with a length cut-off 45 or verystiff force-constants.46 Beyond this bond length potential, one may typicallyinclude a bending energy term to reduce local flexibility. Because the bendingenergy and geometry on this length scale do not derive from chemical hybri-dization, one typically takes the equilibrium bond angle to be 180�. Dihedralenergy terms are generally not included in coarse-grained models. Instead, thechains are treated on mesoscopic scales as freely rotating.

The Bond-Fluctuation Lattice Model

The large-scale structure of polymer chains in a good solvent is that of aself-avoiding random walk (SAW), but in melts it is that of a random walk(RW).11 The large-scale structure of these mathematical models, however, is

Coarse-Grained Models 11

independent of whether one studies them in the continuum or on a lattice,and because of this, MC simulations of lattice models for polymers have along history.

Later in this tutorial we will use results from simulations of the bond-fluctuation lattice model.47–50 This model represents the repeat units of thecoarse-grained polymer not as single vertices on some space lattice but as unitcubes on a simple cubic lattice (see Figure 3 for the three-dimensional version ofthe model). The bonds connecting consecutive monomers are from the class[2,0,0],[2,1,0],[2,1,1],[3,0,0],[2,2,1],[3,1,0], where the square brackets indicateall vectors obtainable from the given vector by lattice symmetry operations.There are a total of 108 bonds with 5 different lengths and 87 different bondangles for this model. The model thus introduces some local conformationalflexibility while retaining the computational efficiency of lattice models forimplementing excluded volume interactions by enforcing a single occupationof each lattice vertex. Intramolecular potentials are chosen as bond lengthand/or bond angle dependent according to the physical problem one wants tomodel. Note that, as in all coarse-grained models, the potentials in the bond-fluctuation model do not correspond physically to bond stretches and valenceangle bending potentials in a chemically realistic polymer chain.

When one implements an MC stochastic dynamics algorithm in thismodel (consisting of random-hopping moves of the monomers by one latticeconstant in a randomly chosen lattice direction), the chosen set of bond vectorsinduces the preservation of chain connectivity as a consequence of excludedvolume alone, which thus allows for efficient simulations. This class of moves

Figure 3 Sketch of the bond-fluctuation lattice model. The monomer units are repre-sented by unit cubes on the simple cubic lattice connected by bonds of varying length.One example of each bond vector class is shown in the sketch.


allows for a physical interpretation of the obtained stochastic dynamics41 thatgenerates Rouse-like motion of the chains12 in the simulation of dense polymermelts.

SIMULATION METHODS

The simulation methods most commonly used for atomistic or coarse-grained molecular models are MD simulations and MC simulations. In MDsimulations, Newton’s equations of motion are integrated to generate a trajec-tory (a history) of the model system. The method can capture all vibrationaland relaxational processes contained in the chosen model Hamiltonian. MC isa stochastic simulation method that can capture relaxational and diffusiveprocesses. MC is commonly used to generate equilibrium configurations foreither sampling of thermodynamic and structural properties or to providestarting configurations for ensuing MD runs that are used to evaluate thedynamics of the model system. In the following discussion, we will examinethe two methods with regard to their two main applications—equilibrationand generation of trajectories for dynamic measurements, respectively.

Monte Carlo Method

The MC method considers the configuration space of a model and gen-erates a discrete-time random walk through configuration space following amaster equation41,51

Pðx; tnÞ ¼ Pðx; tn�1Þ þX

x0Wðx0 ! xÞPðx0; tn�1Þ �

X

x0Wðx! x0ÞPðx; tn�1Þ

½18�

Here x; x0 denote two configurations of the system (specified, for instance, bythe set of coordinates of all atoms f~rng or the position of one chain end for allchains and all bond lengths, bond angles, and torsion angles f~ra1; lai ; y

aj ;f

akg,

where a ¼ 1; . . . M runs over all chains and the indices i; j; k run over all inter-nal degrees of freedom of one chain). The transition rates Wðx! x0Þ are cho-sen to fulfill the detailed balance condition

Wðx0 ! xÞPeqðx0Þ ¼Wðx! x0ÞPeqðxÞ ½19�

which ensures an equal probability flow from x0 to x as in the reverse directionin equilibrium. Here

PeqðxÞ ¼1

Zexpf�bHðxÞg ½20�

Simulation Methods 13

where b ¼ 1=kBT, HðxÞ is the Hamiltonian of the system, and Z is the cano-nical partition function

Z ¼X

x

expf�bHðxÞg ½21�

Equation [19] ensures that the thermodynamic equilibrium distribution of Eq.[20] is the stationary (long-time) limit of the Markov chain generated byEq. [18]. It does not specify the transition rates uniquely, however. Let us writethem in the following way:

Wðx! x0Þ ¼W0ðx! x0ÞWTðx! x0Þ ½22�

where W0 is the probability suggesting x0 as the next state, i.e., to suggest acertain MC move, and WT is the thermal acceptance probability chosen to ful-fill Eq. [19]. This requires the suggestion probabilities to be reversible

W0ðx! x0Þ ¼W0ðx0 ! xÞ ½23�

Only a few choices for WT exist in the literature, i.e., Metropolis rates,Glauber rates, or heat-bath,51 but there is an unlimited variety of possiblechoices for W0, and this is the great advantage of the MC method. Onlysome choices for W0 result in physically reasonable dynamics (in general,local moves like selecting a monomer at random and then moving it intoa randomly chosen direction by a small distance), but all reversible choiceslead to the correct equilibrium distribution of states. One can thereforeinvent MC moves targeted at overcoming the main physical barriers leadingto slow equilibration of a model system.

The two main sources for slow relaxation in polymers are entanglementeffects and the glass transition. The first is entropic in origin, whereas thesecond—at least in chemically realistic polymer models—is primarilyenthalpic. We write the largest relaxation time in the melt as

tlðT;NÞ ¼ tmesðTÞNx ½24�

where tmes is a mesoscopic time scale. The chain length dependence crossesover from x ¼ 2 for Rouse behavior to x ¼ 3:4 for repeating chains.12 Everysimulation method that performs configuration changes typical for the meso-scopic time scale tmes, i.e., local rearrangements, leads to a relaxation of thelarge-scale structure of the polymer chains in the melt only after O(Nx) suchconfiguration changes. This in turn quickly limits the range of chain lengthsone can treat in thermodynamic equilibrium. To circumvent this problem,


one has to use advanced MC techniques that implement global configurationalchanges within a single Monte Carlo step.

A class of these advanced MC techniques consists of the so-called ‘‘con-nectivity altering moves’’ like the cooperative motion algorithm52 and theend-bridging algorithm53–55 and its newest variant, the double-bridgingalgorithm,56,57 which is sketched in Figure 4. The latter two algorithmshave been developed with chemically realistic polymer models in mind,and we will now briefly discuss the concepts behind, and properties of, thesealgorithms.

In the original end-bridging algorithm,53–55 an end monomer i of onechain in the melt attacks a backbone atom j of another chain that is sufficientlyclose in proximity and tries to initiate a change in connectivity of the twoinvolved chains by forming a trimer bridge to this backbone atom. In the eventof a successful bridging, the attacking chain grows by a part that is cut off ofthe attacked chain, and the attacked chain shrinks by a corresponding amount.This description already exhibits the main drawbacks of the algorithm: It gen-erates polydisperse polymer melts, and it needs a sufficient number of chainends to be efficient. It was found empirically that the efficiency of the algo-rithm dropped considerably (1) as the stiffness of the chains was increasedand (2) in the presence of chain orientation. The algorithm was nonethelessapplied successfully to polyethylene melts58,59 and cis-1,4 polyisoprenemelts,60,61 for example.

In the double-bridging algorithm, an inner monomer of a chain attacksan inner monomer of another chain (or the same chain) and tries to form a

Figure 4 Sketch of the double-bridging algorithm. Starting from monomer i on thewhite chain, a trimer bridge to monomer j on the black chain is initiated. If theformation of this connection is geometrically possible, then a bridge between monomersi0 and j0 has to be built, as they are four monomers removed from i and j, respectively. Ifboth bridges can be formed, the intermediate monomers are excised. Two new chainswith the same chain length as the original ones are created


trimer bridge (see Figure 4). Simultaneously another bridge is formed betweentwo monomers that are four steps apart from the first two monomers alongthe two chains, thereby generating two new chains with exactly the samelength as the original chains. These requirements understandably put heavygeometric constraints on the configurations of the two chains for which thistype of move is feasible (because only special choices of involved monomersði; jÞ and excised trimers conserve the chain lengths in the move). The trimerbridge is made of three monomers (atoms) connected by bonds of fixed lengthsl making a fixed angle, which is chosen to be the maximum value of the bond-angle distribution of the model. Whenever the monomers i and j are at a dis-tance less than the maximum bridgeable distance of 4l cosððp� ymaxÞ=2Þ, thisgeometric problem is in principle solvable.

Connectivity changing algorithms are especially efficient in decorrelatinglarge-scale structure in the melt. These algorithms are typically augmented bylocal moves and reptation moves (a randomly chosen end monomer of a chainis cut off and reattached to the opposite end of the same chain with a randomorientation) to equilibrate the local structure.41 An alternative method forovercoming the entropic slowdown in a polymer melt caused by packingand connectivity is the so-called ‘‘4d-algorithm.’’ The idea of this algorithm62

is to turn some monomers into ghost particles (alternatively, one can think ofremoving some particles into the fourth dimension, which is similar to deso-rbing and readsorbing particles from a two-dimensional film into the thirddimension) and then forcing those particles back into the three-dimensionalstructure by applying an external field in an extended ensemble simulation.The algorithm is similar to other suggestions to reduce the packingeffects.63–65 So far it has been tested on the structural relaxation of a collapsedpolymer globule where the connectivity of the chains and the high densityinside the globule lead to a dramatic increase in the structural relaxationtime of the globule.

True equilibration in a polymer melt is only reached when the largestlength scales (e.g., the end-to-end orientation of the chains) are decorrelated.The connectivity altering moves overcome the chain length dependence of thistime scale, and the extended ensemble simulations overcome some packingeffects influencing the prefactor, but we have not yet discussed any methodthat can overcome completely the slowing down in the prefactor ofEq. [24], i.e., the deceleration of the structural relaxation accompanying theglass transition. No efficient MC algorithm exists yet to overcome the slow-down of structural relaxation connected with the glass transition! All MCmoves employed thus far share the same fate as MD simulations, which facean increase of relaxation time (that is, the simulation time needed) by 14orders of magnitude on approaching Tg. Because this range of relaxation timesis not be covered by modern computing machinery, one is limited to follow theglass transition in equilibrium over about 3-4 orders of magnitude in therelaxation time.


Molecular Dynamics Method

MD involves integrating Newton’s equation of motion, which we writein the Hamiltonian form

d~ri

dt¼~vi

~vi

dt¼~Fi=mi

½25�

where mi is the mass of particle i, ~ri is its position, ~vi is its velocity, and~Fi ¼ �rV is the force acting on it, where V is the force field in the Hamiltonian.Because of its conceptual simplicity, the MD method is contained in almost allcommercial simulation packages and is therefore widely used. To use it correctlyand efficiently, however, requires having some knowledge about the integratorsemployed and the strengths and limitations of the method.

The numerical solution of Eq. [25] is typically performed using the velo-city Verlet integrator,37,38 which is a second-order symplectic integrator.66

Symplectic integrators conserve phase space volume and are therefore reversi-ble, endowing them with an excellent stability even for relatively large timesteps and making them good at conserving energy along a microcanonical tra-jectory. The time step in the MD integrator is limited by the fastest degrees offreedom in the material being modeled. When we denote with tf a typicalvibrational period of such a fast degree of freedom, the integration step �thas to be of the order of 1=30tf � 1=10tf . The theory of symplectic integratorsis also the starting point to derive multiple time-step integrators,67,68 whichincrease the efficiency of the MD simulation scheme by calculating weakforces less frequently than strong forces. One can also constrain certain fastdegrees of freedom using methods that ensure that constraints are con-served.34,69,70

Between MC and MD methods, Brownian dynamics (sometimes calledstochastic dynamics) methods exist:71

d~ri ¼~vidt

d~vi ¼~Fi

mi� g~vi

!dt þ sd~WiðtÞ

½26�

The d~Wi are Gaussian white noise processes, and their strength s is related tothe kinetic friction g through the fluctuation-dissipation relation.72 Whenderiving integrators for these methods, one has to be careful to take intoaccount the special character of the random forces employed in these simula-tions.73 A variant of the velocity Verlet method, including a stochasticdynamics treatment of constraints, can be found in Ref. 74. The stochastic


simulation methods introduce an additional, external dissipation mechanisminto the simulation and that alters the native dynamics of the model systemunder study on time scales larger than the typical 1=g time scale for this exter-nal dissipation mechanism.

This is also true for MD simulations in ensembles other than the micro-canonical ensemble. The Nose–Hoover method75,76 for MD simulations in thecanonical ensemble introduces a time-dependent friction gNH as an additionaldynamical variable. Because its average value along the trajectory is small,hgNHi � 1, it affects only very slow processes. This is not true, however, forMD simulations in the NpT ensemble, where the additional barostatdynamics77 strongly changes the intrinsic dynamics of the system. The safestapproach is to work in the microcanonical ensemble, if possible. Starting froman equilibrated configuration, the MD method is used to generate a trajectoryof the model system, and one typically stores configurations along the trajec-tory for a postsimulation analysis of the relaxation processes.

THERMODYNAMIC PROPERTIES

In this section we will discuss two approaches commonly used by scien-tists to study the glass transition in polymers when they determine the tem-perature dependence of thermodynamic properties. These properties includethe specific volume and the specific entropy. As discussed in the Introduction,the break in the temperature dependence of the specific volume served as oneof the first experimental measures of the glass transition.4 In the experiment byKovacs, a very careful study was performed of the cooling rate dependence ofthe glass transition temperature. The break in the temperature dependence ofthe specific volume signifies that the system’s internal relaxation times havereached the time scale of the experiment (inverse cooling rate), at which point,the system is no longer in equilibrium. With simulations one can also use thetemperature dependence of other properties. These properties inlcude theinternal energy or the size of the polymer chains as determined fromthe radius of gyration of the chains,78 both of which are readily accessiblevia computation.

When using an atomic scale MD simulation of this cooling process, theintegration time step dt is typically as small as 1 fs (10�15 s), and accordingly,even very long runs (on the order of 107 or 108 time steps) will not exceed atime interval of 100 ns. It is not even clear that on such short time scales thetrue physical traits of the glass transition can emerge fully. The ‘‘glass transi-tion’’ that was claimed to be observed in an atomistic model for polyethylenechains from simulation time periods less than 1 ns79,80 implied that a freezingof individual jumps between the energy minima of the torsional potential ofthe chains took place, rather than a more collective behavior (‘‘cooperativelyrearranging units’’81) that might occur in the system at somewhat lower


temperatures and on much longer time scales. Even if one takes the view thatthe only influence the short-time window (ns) of MD simulation has is to shiftthe apparent glass transition temperature Tg (MD) upward in comparisonwith that of experiment Tg (expt), one still has to assess the amount of thisupward shift.

Practitioners of MD simulation studying glass forming fluids oftencompare MD results on glass transition temperatures directly with experimen-tal data,82–84 ignoring this systematic difference between Tg (MD) and Tg

(expt). For ‘‘strong’’ glass formers such as molten SiO2, this issue has beenstudied carefully.85 A very strong dependence of Tg (MD) on the coolingrate � was found (which in the case of SiO2 was in the range 1012 K=s <� < 1015 K/s, which is many orders of magnitude higher than in the experi-ments.86) The computed Tg (MD) differed from Tg (expt) 1450 K bymore than 1000 K! Because of the much steeper variation of the structuralrelaxation time t with T near Tg (expt), one does not expect such dramaticeffects for fragile glass formers like most polymers, and indeed for simula-tions,82–84 it was found that Tg (MD) agreed with Tg (exp) reasonably well.

One reason for ignoring the cooling rate dependence of the simulated Tg

in these publications is because MD simulations of chemically realistic modelsare generally too time consuming for a systematic study of different coolingrates (for an exception, see Ref. 87), especially when one takes into accountthat for each cooling rate one must average over several independent coolingruns for each rate as will be described later. To evaluate cooling rate depen-dence, one therefore best uses a coarse-grained model, as was done for thebead-spring model suggested in Refs. 88 and 89. In the latter paper, the melt-ing temperature of the bead-spring model was determined to be T ¼ 0:76 inLennard–Jones units. Upon cooling at a fixed rate in a MD simulation atconstant temperature and pressure (NpT) employing a Nose–Hoover thermo-stat75,76 and an Andersen barostat,77 one observes the temperature-dependentspecific volumes shown in Figure 5. In the top panel, the temperature depen-dence of the specific volume is shown for a cooling rate of � ¼ 52:083 � 10�6.Straight line fits in the melt and in the glassy phase assume a constant thermalexpansion coefficient in both phases. The intersection point between thestraight lines defines the glass transition temperature Tgð�Þ for this coolingrate. In the lower panel, the fit curves are shown for different cooling rates.The melt curve does not depend on the cooling rate, but the glass curvesshow a systematic shift, although, within the uncertainty of the data, theslope of the curves (the thermal expansion coefficient) is independent of thecooling rate also in the glass. It is obvious from the plot that only a small(but systematic) variation in Tgð�Þ exists in the range of cooling rates thatwas accessible in the simulation (3:3 � 10�6 < � < 8:3 � 10�4 in Lennard–Jonesunits).

To interpret the cooling rate dependence of the glass transition tem-perature, one can use the Vogel–Fulcher law discussed in the section on the

Thermodynamic Properties 19

phenomenology of the glass transition. If one assumes that the break in theobserved temperature dependencies occurs when the internal relaxation timeis equal to the time scale of the cooling experiment texp, one obtains

texp ¼ t1 expfEA=ðTgð�Þ � T0Þg ½27�

0.3 0.4 0.5 0.6

T

7.03

7.04

7.05

7.06

7.07

7.08

7.09

ln(V

)

0.3 0.4 0.5 0.6

T

7.03

7.04

7.05

7.06

7.07

7.08

7.09

ln(V

)

8.3 10-4

4.2 10-4

5.2 10-5

6.5 10-6

Figure 5 The upper panel shows the logarithm of the specific volume as a function oftemperature for a cooling rate � ¼ 52:083 � 10�6, with error bars determined from 55independent cooling runs. The lines are fits with a constant expansion coefficient in themelt (continuous line) and glass phase (dashed line), respectively. The lower panel showsthe common fit curve for all cooling rates in the melt and fit curves in the glass for fourcooling rates given in the legend.


In a stepwise cooling experiment, texp is equal to the time spent at every tem-perature step

texp ¼ �T=� ½28�

so that one obtains

Tgð�Þ ¼ T0 � EA= lnð�t1=�TÞ ½29�

Applying this prediction to the cooling rate dependence of a break points inthe specific volume curves, one obtains a Vogel–Fulcher temperature ofT0 ¼ 0:35 that agrees well with that determined from the temperature depen-dence of the diffusion constant in this model, which is TD

0 ¼ 0:32.From these results obtained from MD simulations with a coarse-grained

model, the following picture emerges. When one cools down a model system ina computer simulation, the break in the temperature dependence of the specificvolume indicates the temperature at which the time scale of those internalrelaxation processes involved in volume relaxation equals the time scale ofthe cooling process. In strong glass formers, where the typical time scales ata high temperature increase in an Arrhenius fashion with an activation energythat is much higher than for the fragile glass formers, this means that thesystem falls out of equilibrium on the time scales accessible in an MD simula-tion at temperatures that are much higher than the experimental glass transi-tion temperature and that consequently we obtain a very bad estimate for thistemperature. For fragile glass formers, in contrast, the high-temperatureincrease of relaxation times is in general slow, thus often allowing the simula-tions to approach more closely to the experimental Tg before falling out ofequilibrium. As a result, the glass transition temperatures can be in reasonableagreement with the experimental data. For polymers, where a larger high-temperature activation energy for volume relaxation exists, arising from acoupling to conformational rearrangements involving activated jumps overlarge dihedral barriers, for example, one anticipates the cooling method tobe incapable of locating the experimentally relevant Tg on typical simulationtime scales.

This reasoning also means that we were not really describing a thermo-dynamic measurement of the glass transition in a polymer melt but instead amacroscopic determination of the temperature dependence of volume-relatedinternal relaxation processes, i.e., a dynamic measurement in the disguise of athermodynamic measurement.

Let us now turn to a discussion of the relation of the temperaturedependence of the polymer melt’s configurational entropy with its glass tran-sition and address the famous paradox of the Kauzmann temperature ofglass-forming systems.90 It had been found experimentally that the excessentropy of super-cooled liquids, compared with the crystalline state, seemed


to vanish when extrapolated to low temperatures. The extrapolated tempera-ture of vanishing excess entropy, the so-called Kauzmann temperature TK,is generally in close agreement with the extrapolated Vogel–Fulcher tempera-ture T0 derived from the temperature dependence of relaxation times. Atheoretical derivation of the Kauzmann temperature for polymeric glassformers was given by Gibbs and Di Marzio.91 Consider the canonical parti-tion function of K polymer chains of chain length N in a volume V,

Z ¼X

E

�ðE;K;N;VÞ expð�E=kBTÞ ½30�

where E is the internal energy of the system and � is the microcanonical parti-tion function (i.e., the total number of states). For simplicity, a lattice with Msites is considered with

M ¼ L3

8¼ ðKN þHÞ ½31�

with H being the number of vacant sites (holes). Equation [31] is valid for asimulation of the glass transition in the bond-fluctuation lattice model whereeach repeat unit occupies the eight lattice vertices of a unit cube. The glasstransition in this model was studied by employing a Hamiltonian that singledout the bonds of the class [3,0,0] giving them zero energy92

H ¼ 0 if b 2 ½3; 0;0�E otherwise

�½32�

The entropy density s is then

s ¼ ðln �Þ=M ½33�

In the thermodynamic limit M!1, one can consider s ¼ sðe; rÞ, wheree ¼ E=M is the internal energy per lattice site and r ¼ KN=M is the monomerdensity. We can also do this as a function of temperature because e can bereplaced by T via the appropriate Legendre transformation.

Variants of an approximate calculation of the configurational entropy oflattice chains have been developed by Flory,93 Gibbs and Di Marzio,91 andMilchev.94 All three treatments write � as a product of an intrachain (�intra)contribution and an interchain (�inter) contribution

� ¼ �intra�inter ½34�

In Flory’s original treatment, �intra accounts for the increase of the chain stiff-ness when the temperature is lowered. Flory93 described this chain stiffening by


an energy e if two consecutive bonds along a chain are not collinear, whereas noenergy is assigned if they are collinear. In essence, this calculation relies on thefact that one has a two-level system for the internal degrees of freedom as wasrealized in the choice of Hamiltonian in Eq. [32].

If we denote by f the probability of finding a bond in the excited state, weget for the intramolecular part of the partition function, neglecting excludedvolume effects but assuming a nonreversal random walk, Eq. [35]

�intra ¼KðN � 1ÞfKðN � 1Þ

� �1

z� 1

� �ð1�f ÞKðN�1Þ z� 2

z� 1

� �fKðN�1Þ½35�

where z is an effective coordination number in the melt. The last term in thisequation is obviously correct for the original Flory model where z� 1 possi-bilities exist for the next bond, only one of which is straight and does not carryan energy penalty. For the bond-fluctuation model with the Hamiltonian givenby Eq. [32], 1 in 12 bonds does not carry an energy penalty. Furthermore, theeffective coordination number in the melt is around 12, so again one canassume that one of the z� 1 neighbor bonds is in the ground state.

The treatments of Flory,93 Gibbs and Di Marzio,91 and Milchev94 differin the way they calculate the second factor �inter. This microcanonical parti-tion function describes the number of ways in which the K chains can be puton the lattice,

�inter ¼ 2�Kð1=K!ÞYK�1

k¼0

nkþ1 ½36�

with nkþ1 being the total number of configurations of the ðkþ 1Þth chain ifthere are already k chains on the lattice that can be approximated by

nkþ1 ðM� kNÞNN�1empty zðz� 1ÞN�2 ½37�

Here, M� kN is the number of empty sites after k chains have been placed onthe lattice and constitutes the number of potential starting points for theðkþ 1Þth chain. The factor zðz� 1ÞN�2 represents the number of possibilitiesto place the remaining N � 1 monomers of the chain after the first monomerhas been placed, forbidding only the immediate back-folding of the walk. Thefactor NN�1

empty, which for the bond-fluctuation model counts the number ofempty unit cubes, accounts approximately for the chains being self-avoidingand mutually avoiding and is approximated in the three approaches as inEqs. [38]–[40]:

Nempty ¼ 1� kN=M ðFloryÞ ½38�


or

Nempty ¼ ð1� kN=MÞ=½1� kðN � 1ÞMz=2� ðGibbs--DiMarzioÞ ½39�

or

Nempty ¼ ð1� kN=MÞ=ð1� k=KÞ ðMilchevÞ ½40�

The idea behind these corrections is to recognize that not all empty lattice sitescan serve as starting points for the new polymer; only those lying outside of thevolume already consumed by the other k chains can be used. Unfortunately,neither Eq. [37] nor the expressions for Nempty can be justified with mathema-tical rigor. For both the Flory93 and the Gibbs and Di Marzio91 approxima-tions, the entropy at low temperatures is negative (in the limit N !1 andr! 1)

sðT ! 0Þ ¼ �1 ðFloryÞ ½41�

sðT ! 0Þ ¼ z

2� 1

� �ln 1� 2

z

�< 0 ðGibbs--Di MarzioÞ ½42�

�

whereas Milchev’s94 entropy remains non-negative (sðT ! 0Þ ¼ 0). A compre-hensive account of these different mean-field-like theories has been given byWittmann.95

To test these theoretical approaches in a computer simulation, one needsto proceed in several steps. In the first step, one determines the entropy permonomer in the simulation by measuring the energy per monomer eðr;TÞand by calculating the free energy per monomer through thermodynamic inte-gration of the excess chemical potential96

sðr;TÞ ¼ eðr;TÞT

� rN

lnrN� 1� ln

ZpðN;TÞN

� �� 1

N

ðr

0

mexðr0;TÞdr0 ½43�

where ZpðN;TÞ is the partition function of a single chain of length N. Theexcess chemical potential can be measured according to

mexðr;TÞ ¼ �T ln pinsðr;TÞ ½44�

by evaluating the insertion probability for a chain of length N into a solution atdensity r.97,98 In the second step, one measures the fraction of bonds in theexcited state f, the effective coordination number z (the number of monomersaround a given monomer in the melt that are within a distance given by the max-imum bond length), and the number of holes H (the fraction of the empty latticesites where one can put another monomer of the bond-fluctuation model)


and inserts these temperature-dependent quantities into Eqs. [38]–[40] for thespecific entropy. A comparison between the three theories and computer simu-lation is shown in Figure 6.

The theories of Flory and Gibss–Di Marzio result in practically identicalpredictions, both leading to a Kauzmann paradox (negative excess entropy)around T ¼ 0:18, which is a temperature in the vicinity of the Vogel–Fulchertemperature, T0 � 0:13, determined for this model. Both theories, however,strongly underestimate the value of the configurational entropy, which isalways positive. The reason is that an underestimation of the intermolecularpart of the partition function exists. This underestimation can be observedwhen comparing the behavior at 1=T ¼ 0 with the simulation data and withthe results of Milchev’s theory, which agrees more closely with the simulationdata and stays positive throughout the whole temperature range. All theoriesreproduce the shape of the simulated curve reasonably well, which explainswhy the Gibbs–Di Marzio theory is so successful in predicting experimentalresults on the glass transition in polymer melts that depend only on derivativesof the specific entropy. From the simulations we conclude, however, that theKauzmann paradox of vanishing excess entropy is an extrapolation artifactand that the theoretical descriptions reproducing this finding are based oninappropriate approximations. These studies therefore have not revealed anyevidence for a phase transition underlying the glass transition in polymermelts.

The strong reduction in specific entropy that is observable in experimentas well as in simulation, i.e., the reduction in configuration space available to

0.0 1.0 2.0 3.0 4.0 5.0 6.01/T

0.00

0.05

0.10

0.15

0.20

0.25en

trop

y

Flory

Gibbs–DiMarzioMilchevSimulation

Figure 6 Entropy per monomer in the bond-fluctuation model as a function of inversetemperature. The results from the simulation (filled circles) are compared with thetheoretical predictions discussed in the text.


the chains in the melt, has been linked by Adam and Gibbs [81] to the slowingdown of the dynamics of the system. In the Adam–Gibbs theory, the center ofmass self-diffusion coefficient of the chains is related to the entropy by

DðTÞ ¼ Dð1Þ exp � A

TSðTÞ

� ½45�

Using the specific entropy determined in the simulations, one can test this the-oretical approach by fitting this expression to the temperature dependence ofD observed in the simulations. It has been concluded that the Adam–Gibbstheory cannot predict the temperature dependence of the dynamics from thethermodynamic information contained in the temperature dependence of theentropy.92

In the next sections we will focus on analyzing the dynamics of super-cooled liquids in more detail and discuss our findings in terms of the mode-coupling theory of the glass transition, which is a liquid state theory thatpredicts the dynamics from the structural properties of the liquid.

DYNAMICS IN SUPER-COOLED POLYMER MELTS

It is not easy to obtain a crystalline polymeric material. In order to crys-tallize, a homopolymer chain built of simple regular building blocks like, e.g.,a bead-spring polymer model, can arrange the beads on some regular cubiclattice and have the chains run along one of the lattice directions.89 To dothis, the bead-spring chain must change its random-coil-like state into astretched out state. In real polymer chains, the repeat units will attempt toobtain locally ground state conformations given by the dihedral conformerenergies. These confomers may result in non-space-filling structures. Addition-ally there is polydispersity or chemical randomness, as in the 1,4-polybuta-diene (PB) example that we will be discuss in the section on ‘‘Dynamics in1,4-Polybutadiene.’’

Polymers, therefore, in principle, should be good candidates for (quasi-)equilibrium theories of super-cooled liquids. A liquid state theory that candescribe the onset of slowdown in super-cooled liquids successfully is themode coupling theory (MCT).99–101 It is a microscopic approach to the glasstransition starting from the observation of the freezing-in of the structuralrelaxation in the glass transition. The theory assumes density fluctuations tobe the dominating slow variables in glass-forming systems. Although the the-ory was formulated originally for simple (monatomic) fluids only, it is believedto be of much wider applicability and it has been applied to interpretexperiments on the polymer glass transition.

Starting from the Liouville equation as the fundamental microscopicevolution equation for the dynamics of all phase-space variables, MCT uses


well-established projection operator techniques that are used for eliminatingthe fast variables to arrive at an equation for the correlation functions of den-sity fluctuations in Fourier space, i.e., the intermediate scattering functions.

q2

qt2Sðq; tÞ þ �2

qSðq; tÞ þ �qqqt

Sðq; tÞ þ �2q

ðt

0

mqðt � t0Þ qqt0

Sðq; t0Þdt0 ¼ 0 ½46�

In this generalized oscillator equation, the frequency �q is related to the restor-ing force acting on a particle and �q is a friction constant. The key quantity ofthe theory is the memory kernel mqðt � t0Þ, which involves higher order corre-lation functions and hence needs to be approximated. The memory kernel isexpanded as a power series in terms of Sðq; tÞ

mqðtÞ ¼Xi0

i¼1

1

i!

X

k1...ki

VðiÞðq; k1; . . . ; kiÞSðk1; tÞ . . . Sðki; tÞ ½47�

The coefficients VðiÞ of this mode-coupling functional are the basic controlparameters of this idealized version of MCT. One sees that Eqs. [46] and[47] amount to a set of nonlinear equations for the correlators Sðq; tÞ thatmust be solved self-consistently.

The basic qualitative prediction of MCT is that, upon lowering the tem-perature or increasing the density of a melt, one observes a separation of timescales between the microscopic dynamics and the structural relaxation leadingto a two-step decay of all relaxation functions. One imagines the molecules tobe trapped within a cage formed by their neighbors for some time spanbetween the short time dynamics and the large-scale structural relaxationthat comes about when particles leave their cages. The long-time behavior isthe structural (or a-) relaxation and the plateau regime occuring betweenvibrational dynamics, and this a-relaxation is termed the b-regime in the the-ory. For a well-developed intermediate plateau regime between microscopicand structural relaxation, MCT predicts solutions of the type

Sðq; tÞ ¼ f cq þ hqGðt=t0;sÞ ½48�

in a time window t0 � t� t0s, where f cq is the nonergodicity parameter, hq is

some wave-vector dependent amplitude, G is a q-independent scaling functionof time, and t0 is the microscopic scale. The separation parameter s / 1� T

Tc

measures the distance from the singularity representing the ‘‘ideal’’ glass tran-sition. For s 0, one has

limt!1Sðq; tÞ ¼ f cq þ hq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis=ð1� lÞ

pþOðsÞ ½49�

Dynamics in Super-Cooled Polymer Melts 27

where the parameter lðl < 1Þ is called the ‘‘exponent parameter.’’ For s < 0,on the other hand, one has limt!1fqðtÞ ¼ 0. For s! 0, the functionGðt=t0;sÞ can be linked to a universal correlation function g�ðt=tsÞ, where‘‘þ’’ indicates s > 0 and ‘‘�’’ indicates s < 0

Gðt=t0;sÞ ¼ jsj1=2g�ðt=tsÞ; s > 0ð< 0Þ ½50�

where

ts ¼ t0jsj�1=2a ½51�

and the exponent a is related to l by

l ¼ ½�ð1� aÞ�2=�ð1� 2aÞ ½52�

For short times, one has a power law decay

g�ðt=tsÞ ¼t

ts

�a

� A1t

ts

a

þ . . . ½53�

where A1 is some amplitude. For large t=ts, gþðt=ts !1Þ ¼ 1=ffiffiffiffiffiffiffiffiffiffiffiffi1� lp

, con-sistent with Eq. [49], the correlator approaches structural arrest. The liquidphase solution, on the other hand, exhibits another power law, the so-calledvon Schweidler law,

g�t

ts

� �¼ �B

t

ts

b

;t

ts� 1

� �½54�

where B is another amplitude, and the von Schweidler exponent b is related tol as

½�ð1þ bÞ�2=�ð1þ 2bÞ ¼ l ½55�

One can rewrite the von Schweidler law using Eqs. [48], [50]–[52], [54], and[55] as follows:

Sðq; tÞ ¼ f cq � hqBðt=t0sÞ

b ½56�

where t0s is a characteristic time scale that diverges as the ideal glass transitionis approached from above,

t0s ¼ t0jsj�g; g ¼ 1=ð2aÞ þ 1=ð2bÞ ½57�


This time t0s actually is the maximum time for which Eq. [48] is valid,t0 � t� t0s. The exponent g characterizes the behavior of the a-relaxationor structural relaxation. Typical a time scales diverge as

ta � ðT � TcÞ�g ½58�

As indicated, the power law approximations to the b-correlator describedabove are only valid asymptotically for s! 0, but corrections to these predic-tions have been worked out.102,103 More important, however, is the assump-tion of the idealized MCT that density fluctuations are the only slow variables.This assumption breaks down close to Tc. The MCT has been augmented bycoupling to mass currents, which are sometimes termed ‘‘inclusion of hoppingprocesses,’’ but the extension of the theory to temperatures below Tc or evendown to Tg has not yet been successful.101 Also, the theory is often not appliedto experimental density fluctuations directly (observed by neutron scattering)but instead to dielectric relaxation or to NMR experiments. These latter tech-niques probe reorientational motion of anisotropic molecules, whereas theMCT equation describes a scalar quantity. Using MCT results to comparewith dielectric or NMR experiments thus forces one to assume a direct cou-pling of orientational correlations with density fluctuations exists. The differ-ent orientational correlation functions and the question to what extent theydirectly couple to the density fluctuations have been considered in extensionsto the standard MCT picture.104–108

Of the available experimental techniques, the various neutron scatteringmethods most directly measure structural relaxation. Like simulation techni-ques, however, their dynamic range is limited and several experimental setupshave to be combined to obtain information on polymer relaxation from thepicosecond scale up to the longest time accessible in neutron spin echo experi-ments (� 100 ns depending on momentum transfer), with all the experimentalcorrection and normalization issues involved in matching results from differ-ent experiments. To our knowledge, the reconstruction of a single relaxationcurve Sðq; tÞ out of experimental information for the different frequency andtime windows has not yet been tried. Therefore, simulation studies of the glasstransition still provide us with the most detailed information of the structuralrelaxation processes.109

Before we examine in more detail the dynamics of a super-cooled melt ofcoarse-grained chains and of PB chains, respectively, let us first compare thestructure of these two glass-forming systems. Structure is obtained experimen-tally from either the neutron or the X-ray structure factors. The melt (orliquid) structure factor is given as110

SmðqÞ ¼1

M

XM

n;m

bnðqÞbmðqÞhei~q�ð~rn�~rmÞi ½59�


Here the angular brackets indicate thermal as well as isotropic averages, and thesum runs over all M scattering centers in the melt. The quantities bnðqÞ are thescattering form factors of the different scatterers in the sample. For X-ray scat-tering, momentum transfer (q) dependence of the form factor of the electronicclouds must be taken into account. For neutron scattering, the form factorsreduce to q-independent scattering lengths, bnðqÞ ¼ bn. Neutron scattering stu-dies of the melt structure are typically performed on perdeuterated samples, i.e.,where all H atoms have been replaced by D atoms, because for deuterium andcarbon atoms, coherent scattering dominates (they have about the same coher-ent scattering lengths), whereas hydrogen atoms scatter neutrons incoherently.X-ray scattering is most sensitive to the positional correlations of the heavyatoms in the sample with their large associated electron clouds.

Performing neutron scattering not on perdeuterated samples but on a sin-gle deuterated chain in a protonated matrix (or vice versa; both ways providethe same contrast) gives the single-chain structure factor,

SchðqÞ ¼1

N

XN

n;m

bnbmhei~q�ð~rn�~rmÞi ½60�

where now the sum runs only over all monomers of a single chain.When we think of simulations involving bead-spring models, all scat-

terers can be assigned the same scattering lengths [that are absorbed into arbi-trary units for SðqÞ], and for united atom models like the one used for PB, wecan consider scattering from the united atoms in the same way. This simplifiesthe scattering functions of Eqs. [59] and [60] to be

SmðqÞ ¼1

M

XM

n;m

hei~q�ð~rn�~rmÞi ½61�

and

SchðqÞ ¼1

N

XN

n;m

hei~q�ð~rn�~rmÞi ½62�

The structure factors are Fourier transforms of radial pair-distribution func-tions for the complete melt or the single chain, respectively,

SmðqÞ ¼ 1þ 4prð1

0

r2gmðrÞsinðqrÞ

qrdr ½63�

SchðqÞ ¼ 1þ 4pðN � 1Þð1

0

r2gchðrÞsinðqrÞ

qrdr ½64�


where we already performed the angular average. The pair-distribution func-tions gmðrÞ and gchðrÞmeasure structural correlations directly in real space andare, of course, also observables in the simulations.

We show typical examples for the melt structure factor and for thesingle-chain structure factor in Figure 7. The upper panel is for a chemicallyrealistic simulation of PB,111 where the scattering was calculated with the

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

q [Å-1

]

0

0.5

1

1.5

2

2.5

3

3.5

4

S(q

)

Sm(q) T=273 K

Sch(q) T=273 K

0 5 10 15 20q

0.0

0.5

1.0

1.5

2.0

2.5

3.0

S(q

)

Sm

Sch

Figure 7 Comparison of melt structure factor and single-chain structure factor for PB(upper panel, calculated as scattering from the united atoms only) and a bead-springmelt (lower panel, in Lennard–Jones units).


united atoms as scattering centers of unit scattering length. The lower panelis for a simulation of a bead-spring model.88 Recall that the first maximumin the melt structure factor is called the first sharp diffraction peak or amor-phous halo. The position of the amorphous halo for the PB simulationagrees well with experimental data.112,113

We can see from Figure 7 that for momentum transfers larger than about3 A�1 in PB, i.e., starting around the second maximum, one observes only intra-molecular correlations in the melt structure factor112–114 when one considersonly scattering from the united atom centers. The melt structure factor canalways be decomposed into a chain contribution (SchðqÞ) and a contributionthat captures the correlations between distinct melt chains (SmdðqÞ).

SmðqÞ ¼ SchðqÞ þ SmdðqÞ ½65�

where

SmdðqÞ ¼ 4prð1

0

r2ðgmdðrÞ � 1Þ sinðqrÞqr

dr ½66�

contains only scattering contributions from scattering centers belonging to dif-ferent chains (gmdðrÞ is the pair correlation function for atoms belonging todifferent chains). Intermolecular scattering is responsible for only half of theintensity of the first sharp diffraction peak. For the intermolecular contribu-tion to the scattering, the position of the amorphous halo is given approxi-mately by 2p=d, where d is the typical intermolecular distance betweenscattering centers.

The behavior of the bead-spring model is different from that of PB. Themelt structure factor and the single-chain structure factor depicted in Figure 7only start to agree at the third peak in the melt structure factor. For smallermomentum transfer, they oscillate with the same wavelength but with a phaseshift. The intramolecular structure factor has a minimum preceding the amor-phous halo and a maximum shifted slightly with respect to, but still within, theamorphous halo. For the chemically realistic united atom chain, we observe ashoulder in the intramolecular structure factor at the position of the amor-phous halo and a first minimum where the melt structure factor also has itsfirst minimum. The shoulder tells us that an intramolecular correlation existsin the PB chain on a scale given by the typical intermolecular distance of about4–5 A, which agrees with the size of a repeat unit comprising the chain. Pic-torially, one can think of the bead-spring chain having the local structure of apearl necklace with the beads touching each other, whereas the PB chain con-sists of overlapping spheres with a distance between their centers that isroughly a third of their diameter. The local packing in a hydrocarbon meltlike PB, therefore, resembles more the packing of spaghetti than of billiardballs.


This specific local packing can give rise to scattering behavior that mightbe puzzling at first glance. When one imagines the packing in a polymer meltto be that of billiard balls, one would predict that upon increasing the pressurea better defined packing will result in sharper radial distribution functions and,consequently, in a sharpening and increase in height of the amorphous halo.Moreover, that halo would move to large q because of the overall compressionof the melt. However, a series of experiments on the structure and dynamics ofpolymers under pressure115–118 has been reported, showing a very differentbehavior of the first sharp diffraction peak: It shifted to larger momentumtransfer values as expected, but it simultaneously broadened and decreasedin height. This behavior has been reproduced in a simulation of a chemicallyrealistic model of PB119 under pressure as shown in Figure 8.

To understand the experimental behavior of PB, one has to take intoaccount the fact that the scattering was performed on a perdeuterated sampleand that carbon and deuterium have about the same coherent scatteringlength. Therefore, instead of having one melt structure factor one must actu-ally consider three partial structure factors, SCC, SDD, and SCD, that areweighted by the appropriate combination of scattering lengths (see Ref.110). The partial structure factor SCC is the one we used in Figure 7. To cal-culate the structure factor shown in Figure 8, in contrast, we used the trick of

0 1 2 3 4 5

q [Å-1

]

0

0.5

1

1.5

2

S(q

)

p=1 atmp=2500 atmp=27000 atm

Figure 8 Behavior of the first sharp diffraction peak of PB with experimental scatteringlengths for carbon and deuterium. The deuterium atoms are placed at their mechanicalequilibrium positions determined by the positions of the united atom centers and theequilibrium CH bond length and HCH and HCC bond angles along a united atom MDtrajectory. With increasing pressure (values given in the legend, simulation performed atT ¼ 293 K), the first sharp diffraction peak shifts to larger q as expected butunexpectedly decreases in height.


reinserting deuterium atoms into a time series of stored united atom configura-tions that have been sampled along the MD trajectory. By knowing the equili-brium CH bond length and the equilibrium HCH and HCC bond angles, thehydrogen (deuterium) positions can be uniquely determined from the back-bone configuration of the united atom polymer chain.120,121

Knowing all the partial structure factors, we can conclude that the unex-pected behavior of the scattering function has been induced by the q-depen-dence of the carbon-deuterium cross correlations (which contributepositively when the amorphous halo is located at smaller momentum transfersbut negatively at larger ones) and by a different q-dependence of intramolecu-lar and intermolecular contributions.119 We caution the reader to be carefulwith the interpretation of experimental structure factors, and not just for poly-mers. Given the same molecular packing, neutron scattering on a perdeuter-ated sample, on a partially deuterated sample, or on a protonated sampleand X-ray scattering may yield different experimental structure factors. Onthe other hand, careful analysis of results obtained from different scatteringtechniques and/or isotopic substitution can offer a way to glean informationon partial structure factors from experiment.

We now turn to a characterization of the dynamics in a polymer meltwhere, as it is supercooled, it approaches its glass transition temperature.We begin by looking at the translational dynamics in a bead-spring modeland consider its analysis in terms of MCT.

DYNAMICS IN THE BEAD-SPRING MODEL

Early simulation studies on the structural aspects of the glass transitionin polymer melts were performed using the simple bond-fluctuation latticemodel.122–125 The missing inertial regime of the short-time dynamics andthe discreteness of the lattice, however, limited the information that one couldobtain on structural relaxation using this model. The next simplest polymermodels are hard-sphere chains, studied by Rosche et al.126 using MC simula-tions, and the bead-spring off-lattice model that was studied along an isobarusing MD simulations in an NVT ensemble88 (see also Refs. 127 and 128 forreviews). Using MD as the simulation method has the advantage of capturingthe short-time vibrational dynamics when compared with MC simulations.

Our analysis of the melt dynamics begins by looking at large length andlong time scales where we can assess the temperature dependence of the centerof mass self-diffusion coefficient of the chains. This self-diffusion is measuredin the simulations by monitoring the average mean-squared center of mass dis-placement of all chains and then employing the Einstein relation

DðTÞ ¼ limt!1

hð~RcmðtÞ � ~Rcmð0ÞÞ2i6t

½67�


For a polymer chain, the long time limit in Eq. [67] means that one has to beable to simulate the model system for times on the order of several Rouse timesor, to put it in another way, enough time for the chains to diffuse over a spatialrange a few times their size. This simulation is possible for a bead-spring modeldown to rather low temperatures, but for a chemically realistic model withreasonably long chains, one typically cannot perform such lengthy simulations.Upon super-cooling the bead-spring melts below its crystallization temperature(which is T ¼ 0:76, see the section on thermodynamic properties), and a largedecrease in the self-diffusion coefficient is observed (see Figure 9). The tempera-ture dependence below T ¼ 1 is compatible with a Vogel–Fulcher law with aseemingly vanishing self-diffusion coefficient at T0 ¼ 0:32�0:02. Note,however, that even for this coarse-grained model, which is much easier tosimulate than chemically realistic models, the information on the chain centerof mass diffusion derived from the simulation in a (meta-)stable equilibrium islimited to temperatures above approximately 1:44 T0, which makes thededuction of T0 from these data a risky extrapolation. We will comment onthe MCT fits in this figure later in this chapter.

No crystalline order is visible for the bead-spring model upon cooling tothe frozen-in phase at T ¼ 0:3. The break in the volume-temperature curve(described in the section on thermodynamic information) occurring betweenT ¼ 0:4 and T ¼ 0:45 leads us to expect that the two-step decay describedby MCT should be observable at simulation temperatures above (and closeto) this region. This expectation is borne out in Figure 10, which shows the

0.40 0.60 0.80 1.00T

10-6

10-5

10-4

10-3

10-2

D

VF lawMCT γ=2.09

MCT γ=1.8

Figure 9 Chain center of mass self-diffusion coefficient for the bead-spring model as afunction of temperature (open circles). The full line is a fit with the Vogel–Fulcher law inEq. [3]. The dashed and dotted lines are two fits with a power-law divergence at themode-coupling critical temperature.

Dynamics in the Bead-Spring Model 35

intermediate incoherent scattering function in the bead-spring model for sev-eral values of momentum transfer at T ¼ 0:48.129 The basic length scale ofMCT is the intermolecular distance as given by the position of the amorphoushalo, which is q ¼ 6:9 for the bead-spring model. For q 6:9, we see a well-developed plateau regime in the figure. The amount of decorrelation on themicroscopic time scale increases with q. Also indicated in the figure is thetime scale ts derived from an application of the MCT predictions (Eq. [51])for the b-relaxation regime.

The time scale ts and the amplitudes hq from Eq. [56] are predicted byMCT to show a power law dependence on T � Tc. When one plots ts and theamplitudes hq taken to the inverse of the predicted exponent versus tempera-ture, one can directly find the critical temperature of MCT, T ¼ 0:45, asshown in Figure 11. From the MCT analysis in the b-regime, one also obtainsthe von Schweidler exponent, b ¼ 0:75, and therefore all other exponentsthrough Eqs. [52], [55], and [57]. Another test of MCT, which is suggestedby the form of Eq. (56), is to plot the ratio130,131

RðtÞ ¼fqðtÞ � fqðt0Þfqðt00Þ � fqðt0Þ

½68�

where all times t; t0; t00 are within the plateau region (b-regime, see Figure 12).It follows from Eq. [56] that the function RðtÞ defined in this way has to beindependent of the correlation function that one studies. This so-calledfactorization theorem, i.e., Eq. [68], has been tested in detail for the

10-2

100

102

104

t

0.0

0.2

0.4

0.6

0.8

1.0

φ q(t)

q=1.0q=2.0q=6.9q=9.5q=15

tσ

Figure 10 Intermediate incoherent scattering function for the bead-spring model atT ¼ 0:48 for different values of momentum transfer given in the legend.


bead-spring model132,133 and shown to be valid for many correlators, includ-ing coherent as well as incoherent scattering and the Rouse modes.

The von Schweidler exponent, b ¼ 0:75, obtained from the b-relaxationdetermines the exponent for the a-relaxation to be g ¼ 2:09. This exponentshould be observable in the temperature dependence of the self-diffusion coeffi-cient shown in Figure 9. The dashed line in this figure is a fit with fixed values ofTc ¼ 0:45 and g ¼ 2:09 as determined from the b-relaxation analysis; the dottedline is the best fit at fixed Tc. The quality of the fit can be improved if one allowsthe exponent to differ from the prediction based on the b-relaxation behavior.Actually, a systematic decrease of the best-fit value for the exponent in thea-relaxation temperature dependence with increasing length scale wasobserved.134 The a time scale is also the time scale of the final decay in the scat-tering functions Sðq; tÞ, be it coherent or incoherent scattering. This time scale istypically obtained by fitting a KWW (Eq. [9]) time dependence to the final decayof the correlators. In a finite temperature regime above Tc, the stretching expo-nent b of the KWW functions is independent of temperature. In this temperaturerange, b is momentum transfer dependent with values between 0.65 and 0.75and approaching the value of the von Schweidler exponent for q!1. Inthe temperature window with constant b, the time-temperature superpositionprinciple, which is often used to reconstruct complete time-dependent curvesfrom experimental measurements at different temperatures, is valid. The physi-cal interpretation of the time-temperature superposition is that molecular

0.44 0.46 0.48 0.50 0.52T

0.0

0.2

0.4

0.6

0.8

1.0

t σ–2a

100(

h qs )2 |σ|

tσq=3q=6.9q=9.5

Figure 11 MCT b-scaling for the amplitudes of the von Schweidler laws fitting theplateau decay in the incoherent intermediate scattering function for a q-value smallerthan the position of the amorphous halo, q ¼ 3:0, at the amorphous halo, q ¼ 6:9, andat the first minimum, q ¼ 9:5. Also shown with filled squares is the b time scale. Allquantities are taken to the inverse power of their predicted temperature dependence suchthat linear laws intersecting the abscissa at Tc should result.


relaxation mechanisms are the same within this temperature window. This isgenerally valid only over limited temperature ranges.

The structural relaxation time scale, however, also determines relaxationprocesses on large length scales like, for example, the decay of the Rousemodes.135,136 In Figure 13 we can see that the time scales of the first five Rousemodes follow the predicted a-scaling of relaxation times over a certain tem-perature window above Tc. When one comes too close to Tc, the systemdoes not actually freeze. Instead, other relaxation processes not consideredin the idealized MCT take over. The glassy freezing therefore enters into theRouse picture only through the temperature dependence of the segmental fric-tion �ðTÞ, following the temperature dependence of the a-relaxation timescale.

One also finds that the value of the a-exponent (and consequently allother exponents) does not depend on the thermodynamic path one followsto reach the state-point given by ðTc; rcÞ,134 which conforms to another pre-diction by MCT. In summary, MCT has been found to be consistently applic-able to the glassy slowdown in the bead-spring polymer model over a narrowrange of temperature above Tc. What is the reason underlying the applicabilityof MCT, considering the fact that the theory was developed for simple liquidsand has no connectivity built in? The physics behind the success of MCTin describing the slowdown in bead-spring melts becomes clear when welook at the mean-square displacement master curve. This curve is obtained

10-2

10-1

100

101

102

103

104

t

-5

0

5

Rq(t

)

q=1

q=6.9

q=19

Figure 12 Test of the factorization theorem of MCT for the intermediate coherentscattering function for the bead-spring model and a range of q-values indicated in theFigure. Data taken from Ref. 132 with permission.


by plotting the displacements at all simulated temperatures against DðTÞt,where DðTÞ is the chain center of a mass diffusion coefficient at a given tem-perature, as shown in Figure 14. Also shown in the figure is the same mastercurve constructed for a binary Lennard–Jones fluid.137 For t!1, the data for

Figure 13 Temperature dependence of the time scales for the first five Rouse modes inthe bead-spring model in the vicinity of the MCT Tc.

10-7

10-6

10-5

10-4 10

-310

-210

-110

010

110

2

Dt

10-4

10-2

100

102

g 1(t)

diffusive

t 0.63

6 rsc2 + A (Dt)

0.75

LJ mixture

Rg2

Re2

Figure 14 Master curve generated from mean-square displacements at differenttemperatures, plotting them against the diffusion coefficient at that temperature timestime. Shown are only the envelopes of this procedure for the monomer displacement inthe bead-spring model and for the atom displacement in a binary Lennard–Jones mixture.Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusiveregime for the bead-spring model (� t0:63), the MCT von Schweidler description of theplateau regime, and typical length scales R2

g and R2e of the bead-spring model.


the two models must agree by construction. At very early times, the particlesmove freely, and both models exhibit ballistic motion, followed by the slowdisplacement characteristic for the b-process of MCT. However, where theLennard–Jones fluid directly crosses over from the cage effect to the free diffu-sion, the polymer exhibits an intervening connectivity-dominated regime forlength scales between the bond length, l ¼ 1; and the end-to-end distance. Inthis regime, the observed mean-square displacement increases less quickly(� t0:63) than it does in the MCT description, which is here displayed as theeffective von Schweidler law

g1ðtÞ ¼ 6r2sc þ A1ðDtÞ0:75; ðrsc ¼ 0:087; A1 ¼ 11:86Þ ½69�

before free diffusion sets in. This difference from simple liquids, i.e., the cross-over to Rouse-like motion, has recently been included in mode couplingtheory.138,139

Furthermore, it is important to note that the values of the plateau displa-cement in the bead-spring model as well as in the Lennard–Jones liquid modelare well below one. For the Lennard–Jones liquid, this is the scale of the inter-molecular packing s ¼ 1, but for the bead-spring polymer, this is also the scaleof the bond-length l � s ¼ 1. The packing constraints thus act on a lengthscale much smaller than the bond length in this polymer model; i.e., the mono-mers are caged before they actually feel that they are bonded along the chain.This might also explain why both models have very similar critical tempera-tures (Tc ¼ 0:45 for the bead-spring model vs. Tc ¼ 0:435 for the LJ mixture).

As we discussed in the section on the structural properties of amorphouspolymers, the relative size of the bond length and the Lennard–Jones scale isvery different when comparing coarse-grained models with real polymers orchemically realistic models, which leads to observable differences in the pack-ing. Furthermore, the dynamics in real polymer melts is, to a large extent,determined by the presence of dihedral angle barriers that inhibit free rotation.We will examine the consequences of these differences for the glass transitionin the next section.

DYNAMICS IN 1,4-POLYBUTADIENE

Structural relaxation in glass-forming polymers has been studied formany years using chemically realistic simulations. Most of the early workthat examined incoherent, as well as coherent scattering functions, is moreof a qualitative nature because of the unsatisfactory quality of the force fieldsemployed and the severe limitations on the length of the MD simulationsperformed. Roe studied the slowdown of structural relaxation in a PE-likemodel140,141 as well as for polystyrene.142 More recently Okada et al.143,144


performed MD simulations of cis-1,4-polybutadiene143,144 to identify candi-dates for jump motions in asymmetric double-well potentials. The idea ofjumps in double-well potentials had been used earlier to explain quasi-elasticneutron scattering (QENS) data145,146 on PB. A detailed analysis of mode-coupling predictions was performed by van Zon and de Leeuw for a PB-likemodel147 and a PE-like model148 and by Lyulin et al.149–151 for polystyrene. Inthe work by van Zon and de Leeuw, no quantitative comparison with experi-mental data was possible because of the limitations of the force field qualityand the short runs performed. In the work of Lyulin et al., the simulationsextended to several tens of nanoseconds, but there too no quantitative compar-ison was made with the experiment.

In the works of Lyulin et al. and of van Zon and de Leeuw, MCT wasfound to provide a consistent description of the coherent as well as incoherentintermediate scattering functions over a temperature range between Tc andabout 1:2 Tc. The value of the critical temperature van Zon and de Leeuwobtained for PB was Tc � 162 K, which is about 50 K below the experimentalvalue. Their conclusions mostly agreed with those derived for the bead-springmodel with an important difference:148 The relaxation in real polymers is farmore stretched than in the bead-spring model. This stretching results in a smal-ler value of the von Schweidler exponent (b ¼ 0:46 for the PE-like model vs.b ¼ 0:75 for the bead-spring model) as well as for the KWW stretching expo-nent (b � 0:4 for the PE-like model vs. b � 0:7 for the bead-spring model),which are interrelated by the MCT prediction bq ! b for q!1. The resultfor the KWW exponent for the chemically realistic simulation agrees wellwith typical values found in neutron scattering or dielectric experiments. Incontrast, the result for the bead-spring model is much larger.

We can therefore conclude that differences in the structural relaxationbetween bead-spring and chemically realistic models can be attributed to eitherthe differences in packing that we discussed above or the presence of barriers inthe dihedral potential in atomistic models. To quantify the role of dihedralbarriers in polymer melt dynamics, we now examine high-temperature relaxa-tion in polymer melts.

There has been extensive effort in recent years to use coordinated experi-mental and simulation studies of polymer melts to better understand the con-nection between polymer motion and conformational dynamics. Although noexperimental method directly measures conformational dynamics, severalexperimental probes of molecular motion are spatially local or are sensitiveto local motions in polymers. Coordinated simulation and experimentalstudies of local motion in polymers have been conducted for dielectricrelaxation,152–158 dynamic neutron scattering,157,159–164 and NMR spin-lat-tice relaxation.17,152,165–168 A particularly important outcome of these studiesis the improved understanding of the relationship between the probed motionsof the polymer chains and the underlying conformational dynamics thatleads to observed motions. In the following discussion, we will focus on the

Dynamics in 1,4-Polybutadiene 41

information obtained from NMR experiments that have been used to probelocal reorientational motion.

NMR 13C spin-lattice relaxation times are sensitive to the reorientationaldynamics of 13C–1H vectors. The motion of the attached proton(s) causes fluc-tuations in the magnetic field at the 13C nuclei, which results in decay of theirmagnetization. Although the time scale for the experimentally measured decayof the magnetization of a 13C nucleus in a polymer melt is typically on the orderof seconds, the corresponding decay of the 13C-1H vector autocorrelation func-tion is on the order of nanoseconds, and, hence, is amenable to simulation.

The spin-lattice relaxation time T1 can be determined from simulation byusing the relationship169 of Eq. [70]

1

nT1¼ K½JðoH � oCÞ þ 3JðoCÞ þ 6JðoH þ oCÞ� ½70�

where JðoÞ is the spectral density as a function of angular frequency given by

JðoÞ ¼ 1

2

ð1

�1P2ðtÞ expfiotgdt ½71�

Here n is the number of attached protons at a given carbon atom and oH andoC are the proton and 13C resonance frequencies. The constant K assumes avalue of 2:29� 109 s�2 and 2:42� 109 s�2 for sp3 and sp2 nuclei, respectively.The orientational autocorrelation function is obtained from the simulation tra-jectory using the relationship

P2ðtÞ ¼1

2½3hjjeCHðtÞ � eCHð0Þjj2i � 1� ½72�

where eCHðtÞ is the unit vector along a C–H bond at time t. Equation [72] is anensemble average over all carbon atoms with the same chemical environment.Experimentally, T1 values can be determined for 13C nuclei in various chemi-cal (bonding) environments because of the different chemical shifts of thesenuclei (the resonances one can distinguish in a cis-trans copolymer of polybu-tadiene are shown in Figure 15).

The chemically realistic simulations we are discussing have been per-formed using a united atom representation of PB, which leads to the question:How does one actually measure a CH vector reorientation for such a model?The answer to this question is to use the trick we discussed in the analysis ofthe pressure dependence of the melt structure factor of PB. Hydrogen atomsare placed on the backbone carbons at their mechanical equilibrium positionsfor each structure that has been sampled along the MD trajectory. The CHvector dynamics we are showing in Figure 16 is solely from the backbone reor-ientations of the chain.


We can see that the different positions along the chain show distinct tem-perature-dependent relaxation curves. To further analyze these relaxationfunctions, we must Fourier transform them to determine their spectral density,which is best done employing an analytic representation of the data that

cis trans

cis−trans trans−cis

cis cis

cis−cis cis−cis

trans

trans

trans−trans

trans−trans

Figure 15 Sketch of the local environment along a polybutadiene chain of cis-and trans-conformers. For sp3-hybridized carbon atoms (indicated by the gray spheres), thechemical shift is different when they belong to a cis-monomer than when they belongto a trans-monomer. For sp2-hybridized carbon atoms (shown by black spheres) in acis-monomer, NMR shows a different chemical shift whether they have anothercis-monomer as a neighbor or a trans-monomer as a neighbor, and it is similar for thesp2-hybridized carbon atoms in the trans monomer.

10-2

100

102

104

t [ps]

10-4

10-3

10-2

10-1

100

P2(t

)

cis T=353 Kcis T=293 Kfit functiontrans-trans T=353 Ktrans-trans T=293 K

Figure 16 Second Legendre polynomial of the CH vector autocorrelation function forthe sp3 cis-carbon (dashed lines) and the sp2 carbon in a trans-group next to a trans-group (dashed-dotted lines) for two different temperatures. The fit curves to the cis-correlation functions are a superposition of exponential and stretched exponentialdiscussed in the text.


enables us to extrapolate to ‘‘infinite’’ time. For this process, we fit the data tothe following superposition of exponential and stretched exponential decay

P2ðtÞ ¼ Ae�t=t1 þ ð1� AÞe�ðt=t2Þb ½73�

As one can see in Figure 16, where two fit functions are included, our ansatzfor P2ðtÞ can describe the data very well, except for the sub-picosecond vibra-tionally dynamics, which, however, has a negligible contribution to thespin-lattice relaxation time. From this information, we can then calculatethe T1 times for different positions along the chain.

For PB, comparison of 13C NMR spin-lattice relaxation times and nuclearoverhauser enhancement (NOE) values from simulation and experiment over awide range of melt temperatures revealed excellent agreement.168 A comparisonbetween simulation and experiment for two temperatures for six different reso-nances is shown in Figure 17. Comparing the variation in T1 for carbon atoms indifferent chemical environments with the variation in mean waiting timebetween conformational transitions for the different types of torsions presentin PB (b, cis-allyl and trans-allyl), one concludes that spin-lattice relaxationfor a given nucleus in PB cannot be associated with the dynamics of any parti-cular torsion. Instead, the 13C relaxation occurs as the result of multiple confor-mational events involving several neighboring torsions.168 However, a closecorrespondence was found to torsional autocorrelation times tTOR. The tor-sional autocorrelation time is given by the time integral of the torsional autocor-relation function

�tor ¼hcosfðtÞ cosfð0Þi � hcosfð0Þi2

hcos2 fð0Þi � hcosfð0Þi2½74�

Figure 17 Spin-lattice relaxation times for six resonances along a PB chain. Trans andcis denote sp3 hybridized carbons in the respective monomer type, trans-trans, trans-cis,cis-cis, and cis-trans. They denote sp2 hybridized carbons in a trans-group with a trans-group as neighbor, a trans-group with a cis-group as neighbor, and so on. Open bars arefor simulation, and filled ones for experiment. Values are shown for 273 K (short bars)and 400 K (longer bars).


where fðtÞ is the dihedral angle for a particular torsion at time t and the aver-age is taken over all dihedrals of a given type. The C–H vector correlation timetCH is given as the time integral of P2ðtÞ (see Eq. [73], upon which T1

depends). This correspondence is illustrated in Figure 18.The close correspondence between tTOR and tCH shown in Figure 18 has

also been observed in simulations of other polymer melts.152,165 Interestingly,both the C–H vector and the torsional correlation times exhibit stronger thanexponential slowing with decreasing temperature, whereas the rate of confor-mational transitions exhibits Arrhenius temperature dependence as shown inFigure 18. The divergence of time scales between the torsional correlation timeand the rate of conformational transitions is a first indicator of increasingdynamic heterogeneity with decreasing temperature. We will come back tothis point later on. The T1 values themselves were found to correlate wellwith tCH at higher temperatures as expected for the extreme narrowing regime(NOE is approximately three). However, at lower temperatures, the tempera-ture dependence of T1 corresponds neither to that observed for tCH nor to thatobserved for the mean conformational transition times, which implies that thetemperature dependence of the experimentally measurable T1 values revealsno quantitative information about the dynamics of the underlying conforma-tional motions that lead to spin-lattice relaxation.168

A similar analysis in terms of conformational dynamics can be per-formed as well for the interpretation of neutron scattering data in the pico-second time window159 and dielectric data.156 How do these findings then

Figure 18 Temperature dependence of the C–H vector (selected, filled symbols) andtorsional correlation (open symbols) times for PB from simulation. Also shown is themean waiting time between transitions for the cis-allyl, trans-allyl, and b torsions in PB.The solid lines are VF fits, whereas the dashed lines assume an Arrhenius temperaturedependence.


relate to the interpretation of translational motion and scattering functions insuper-cooled liquids in terms of MCT that was presented above for the bead-spring model?

To separate the contributions of dihedral barriers and packing effects,Krushev and Paul111 compared simulations of PB using the chemically realisticforce field (CRC) with identical calculations in which all torsion energies wereset to zero (FRC¼ freely rotating chain). Because the different conformationalstates in PB are almost isoenergetic and the torsion potentials are highly sym-metric, no discernible influence was detected on either the single chain structurefactor or the liquid structure factor.111 Furthermore, the long-time dynamicswas only rescaled by a change in diffusion coefficient.170 All simulations wereperformed at high temperatures using runs on the order of 100 ns, to ensurethat the results were not influenced by quenching in non-equilibrium structures.

A comparison of the melt structure factor and the single-chain structurefactor of these models shows complete agreement.111 According to mode-coupling theory, both models should then show the same dynamics.Figure 19 reveals that this is not the case: For the FRC model at 273 K, oneobserves a crossover from short time vibrational motion to Rouse-like motion,whereas the CRC model shows a well-defined plateau regime between theshort-time and the long-time behavior at this temperature. This plateau regimeis not present for the CRC model at high temperatures and extends in time asthe temperature is lowered. The physical origin for this separation of time scalesis not the packing as assumed in MCT but the presence of intramolecular bar-riers. The short-time vibrational motions are damped out on a time scale of

10-2

10-1

100

101

102

103

104

105

t [ps]

10-3

10-2

10-1

100

101

102

103

104

∆R2 [Å

2 ]

353K CRC240K CRC273K CRC273K FRC

Figure 19 Mean square monomer displacements using the CRC model of PB at threetemperatures compared with the monomer displacement in an FRC version of thepolymer model. Also indicated is the Rouse-like regime with the subdiffusive t0:61 powerlaw entered after the caging regime (CRC at low T) or after the short time dynamics(FRC and CRC at 353 K).


about 1 ps for all temperatures. Local reorientation needs transitions over thebarriers in the torsion potentials (as discussed in detail earlier). These thermallyactivated processes occur on an average time scale of several picoseconds at hightemperatures. Upon lowering the temperature, the waiting time betweentorsional transitions increases in an Arrhenius fashion (see Figure 18), whichleads to a separation of time scales between vibrational motions and structuralrelaxation. Thus, in polymers, we have a second mechanism for the time scaleseparation between short-time vibrational motion and a-relaxation besides thepacking mechanism considered in MCT.

Packing effects can contribute to the increase in waiting times betweentorsional transitions that require enough thermal energy to overcome the intra-molecular barrier as well as space to accommodate the change in local confor-mation. This may require neighboring chains to move out of the way, which inturn may require those neighbor chains to undergo a torsional transition. Soan intricate interaction exists between the intramolecular energetics and thelocal packing. It is not yet understood how the influence of packing versusthe influcence of torsional barriers balances as a function of temperature.For the melt regime in PB, however, it has been established that the activationenergy for the Arrhenius temperature dependence of the mean waiting timebetween torsional transitions is given by the intramolecular dihedral barriersalone168 (see Figure 18).

A quantitative assessment (by simulations) of the low-temperaturedynamics in the super-cooled melt for a polymer like PB requires ‘‘well-equilibrated’’ starting configurations. For the PB model we discuss here theseconfigurations have been generated using parallel tempering techniques171

combined with very long (several hundred nanoseconds) MD runs. Wesurround ‘‘well equilibrated’’ with quotation marks because, as discussed,one cannot propagate the chemically realistic model chains into the free diffu-sion limit at low temperatures (temperatures approaching Tc). However, thevolume of the model systems can be equilibrated and the runs are morethan long enough to equilibrate local conformational statistics. As we dis-cussed, no relevant temperature dependence of the coil-structure exists inPB, which makes this polymer an ideal model system where one can expectto observe only small effects developing from the time-scale limitations ofchemically realistic simulations.

In the discussion on the dynamics in the bead-spring model, we haveobserved that the position of the amorphous halo marks the relevant locallength scale in the melt structure, and it is also central to the MCT treatmentof the dynamics. The structural relaxation time in the super-cooled melt is bestdefined as the time it takes density correlations of this wave number (i.e., thecoherent intermediate scattering function) to decay. In simulations one typi-cally uses the time it takes Sðq; tÞ to decay to a value of 0.3 (or 0.1 for largerq-values). The temperature dependence of this relaxation time scale, which isshown in Figure 20, provides us with a first assessment of the glass transition


in PB. The temperature dependence of the a-relaxation time can be describedover a large temperature interval by a VF law, as was found when computingthe diffusion coefficient of the bead-spring model. The Vogel–Fulcher tem-perature used in Figure 20 is not an independent fitting parameter; it wasobtained by determining from the simulation the dielectric a process for tem-peratures above 253 K.156 T0 determined in this way agrees with the resultsfrom dielectric experiments172,173 and with dynamic mechanical measure-ments.13 This VF law, however, fails to describe the low-temperature behaviorof the a time scale, which is purely Arrhenius as the dashed fit curve indicatesthat it is actually a superposition of two Arrhenius laws. The low-temperatureArrhenius behavior has an activation energy of 5650 K, which illustrates thegeneral finding that the VF law is a crossover law that can interpolate success-fully between a high- and a low-temperature behavior. Upon approaching theglass transition temperature, which for PB is 178 K, the relaxation time tem-perature dependence becomes Arrhenius-like and the VF law fails.174,175 TheVogel–Fulcher temperature T0, therefore, is an extrapolation artifact similarto the Kauzmann temperature.

In the interval between 198 K and 253 K, the form of the structuralrelaxation does not change114 as is evidenced by the success of the time-temperature superposition shown in Figure 21. One can also see from thisfigure that an additional regime intervenes between the short-time dynamics(first 10% of the decay at the lowest temperatures) and the structural relaxa-tion (last 80% of the decay). We will identify this regime as the MCT b-regime

200 250 300 350 400T [K]

1

100

10000

1e+06

τ α(q=

1.4

Å-1

) [p

s]

incoherent1/3 coherentVF with T

0=127 K

2 Arrhenius laws

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time isdefined as the time it takes for the incoherent (circles) or coherent (squares) intermediatescattering function at a momentum transfer given by the position of the amorphoushalo (q ¼ 1:4A�1) to decay to a value of 0.3. The full line is a fit using a VF law with theVogel–Fulcher temperature T0 fixed to a value obtained from the temperaturedependence of the dielectric a relaxation in PB. The dashed line is a superposition oftwo Arrhenius laws (see text).


later. The duration of this additional regime increases with decreasing tem-perature, and its amplitude increases with decreasing temperature.

We can analyze this intervening time regime as we did for the bead-spring model by fitting it with an extended von Schweidler law

Sðq; tÞ ¼ f cq � hqtb þ hqð2Þt2b � . . . ½75�

The von Schweidler law describes well the decay from the plateau in boththe coherent and the incoherent scattering functions. All correlators for PB canbe fitted with a von Schweidler exponent b ¼ 0:3 (see Figure 22). Like thefindings by van Zon and de Leeuw148 and by Lyulin and Michels,149 the decayis much more stretched for the chemically realistic PB model than for thebead-spring model. Typical values for the stretching exponent in the KWW fitto the a relaxation (which should approach b for q!1) are around 0.5, whichagrees well with experimental values. Using the b fit parameters, we candetermine the critical temperature in a manner that is similar to what we didfor the bead-spring model. The value we obtain this way is Tc ¼ 214� 2K,176 which agrees perfectly with experiment.177 It has so far not been possibleto obtain a value for the von Schweidler exponent experimentally, which we cantherefore predict by these chemically realistic simulations of PB.

Finding that the scattering functions at low temperature are amenable toan MCT description, we are faced with a dilemma. On the one hand, the high-temperature mean-square displacement curves lead us to conclude that dihe-dral barriers constitute a second mechanism for time scale separation insuper-cooled polymer melts besides packing effects. On the other hand, the

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 101t / τα

0

0.2

0.4

0.6

0.8

1

S(q=

1.4

Å-1

,t)

T = 198 KT = 213 KT = 222 KT = 225 KT = 228 KT = 240 KT = 253 KKWW

Figure 21 Coherent intermediate scattering functions at the position of the amorphoushalo versus time scaled by the a time, which is the time it takes the scattering function todecay by 70%. The thick gray line shows that the a-process can be fitted with aKohlrausch–Williams–Watts (KWW) law.


plateau regime in the low-temperature scattering data is perfectly described byMCT. The resolution of this dilemma can be observed in the fact that for PBMCT does not correctly predict the temperature dependence of the a timescale in the vicinity of Tc. The exponent g governing the divergence of the atime scale on approaching Tc is much smaller than that calculated from thevon Schweidler exponent using the exponent relations of MCT.176 The differ-ence between successful description of the b regime and the failure in the aregime can be an indication that the plateau regime is packing dominatedand the structural relaxation is influenced by both mechanisms for time scaleseparation, but at this point in time, our arguments are only speculative. Addi-tional studies of PB models with modified dihedral barriers are under way178

to provide more insight into which mechanism dominates the relaxation pro-cesess in which time regime.

DYNAMIC HETEROGENEITY

It has been clearly demonstrated by experiments as well as by simulationsthat the glass transition phenomenon is associated with an increase in dynamicheterogeneity in the motion of the glass-forming moieties. This heterogeneity isbest observed experimentally at temperatures close to or below Tg, where theheterogeneity is well developed.179–181 The existence of domains of fast- andslow-moving molecules is closely connected with the existence of a characteris-tic length scale measuring correlated behavior and with the temperature depen-dence of this length scale.182 As far as geometrical information can be inferredfrom the experiments (as, for example, in References 183–185), these regions

0.1 101 100 1000 10000 1e+05 1e+06t [ps]

0

0.2

0.4

0.6

0.8

1

S(q=

1.4

Å-1

,t)

T = 198 KT = 213 KT = 222 KT = 225 KT = 228 KT = 240 KT = 253 K

Figure 22 von Schweidler fits (dotted lines) to the plateau decay of the coherentintermediate scattering function in the temperature interval 198–253 K.


seem to be on the order of a few nanometers in size, even at and below Tg, andthere is no indication that this length scale is becoming macroscopic.

Using computer simulations of simple liquids,186,187 it was suggestedthat the non-Gaussianity of the van-Hove correlation functions for tagged par-ticle motion can identify fast-moving particles. Typically, the van Hove func-tion Gsðr; tÞ from the simulation develops a tail when compared with theGaussian approximation GG

s ðr; tÞ having the same second moment, i.e., havingthe same mean-square displacement of the molecules. Defining the fraction ofmolecules with displacements beyond the crossing point r defined byGsðr ; tÞ ¼ GG

s ðr ; tÞ as fast-moving particles yields around 6% of fast parti-cles. This definition of fast-moving and variants thereof have been appliedto identify such particles and an eventual clustering phenomenon from simula-tions of the bead-spring polymer model.188–190

When one studies the clustering properties of these fast particles, onefinds that the clusters are typically very ramified, string-like objects. The aver-age mass of the clusters as a function of time lag from a starting configurationhas a peak in the vicinity of the late b regime, where the non-Gaussianity of thevan Hove function is maximum. At this point in time, the particles break outof their cages. This peak in the average cluster size is a consequence of theidentification of the fast particles as those being faster than predicted by theGaussian behavior. A dynamic correlation length (defined as the weight aver-age mean-squared size of the cluster) increases only from x � 2:5s to x � 3:1son approaching Tc from above, where s is the Lennard–Jones radius of thebead-spring monomers. This small increase is compatible with the experimen-tal finding that the typical size of a domain of fast relaxing molecules attemperatures below Tc is only a few nanometers, which translates intobetween 3 and 10 Lennard–Jones radii.

The new qualitative insight obtained from the cluster analysis is thestring-like character of clusters of fast particles, which means that theseparticles tend to follow each other along their paths of movement. However,only very short scale correlation of this motion exists. There is also noobservable tendency for the bead-spring monomers to move along their chaincontour to take up the position of their bonded neighbor. In contrast, for PB,a clear tendency exists for a monomer to replace its bonded neighbor on theaverage time scale of a torsional transition.191

Torsional transitions combined with the local packing in a melt of chemi-cally realistic chains (see the discussion on structural properties) give rise to a dis-tinct type of motion191 as shown in Figure 23. Here we show isosurfaces of theintrachain distinct part of the van Hove function, that is, the probability that oneparticle of a chain is at the origin at time zero and another particle is at position rat time t. The structure along the r-axis for small t gives the intrachain radial dis-tribution function with distinct peaks created by the C–C bond length, the aver-age C–C–C bond angle, and the torsional isomers. On the average time scale of atorsional transition (around 100 ps), the bonded neighbor moves into the space

Dynamic Heterogeneity 51

that had been occupied by the reference united atom at time zero. That is, the tor-sional transitions lead to a slithering motion of the chain along its contour. Thismotion is also observable in the van Hove self-functions for the attached hydro-gen atoms,192 and it influences the scattering at lower temperatures.164

As mentioned, the measures for dynamic heterogeneity typically applied tothe bead-spring model are similar to those used for simple liquids. For chemi-cally realistic polymer models, however, a much simpler measure of dynamicheterogeneity has been used for many years in this type of simulation. This het-erogeneity is found in transition rates for different chemically identical dihedralsin the melt. This heterogeneity shows up most strikingly in the divergence in therate of conformational transitions for dihedrals, which follow an Arrhenius tem-perature dependence with activation barriers resulting primarily from internalrotation barriers, and the relaxation time for the torsional autocorrelation func-tion154,156,165,168,193,194 shown in Figure 18. Although at high temperatures therelaxation times for torsional autocorrelation functions closely follow the rate ofconformational transitions,165 the former exhibit Vogel–Fulcher-like tempera-ture dependence, whereas the latter stay Arrhenius-like. The torsional autocor-relation times are closely related to the rates of local relaxation, which give riseto experimentally measurable quantities like the NMR spin-lattice relaxationtime, as discussed. The Vogel–Fulcher parameters that one typically finds forthe relaxation time scales for the torsional autocorrelation function agree wellwith those parameters obtained for dielectric or magnetic relaxation fromexperiment.168

The divergence between the rate of conformational transitions and thedecay of the torsional autocorrelation functions (and hence local relaxations)

Figure 23 Isosurface of the intrachain distinct part of the van Hove function projectedonto the time-distance plane. For t! 0, one observes the intrachain pair correlationfunction along the radial axes. On the average time scale of a torsional transition, abonded neighbor moves into the position that the center particle occupied at time zero;i.e., the chain slithers along its contour.


in amorphous polymers may lie in the increasingly heterogeneous nature of con-formational transitions with decreasing temperature, which is an observationthat has now been confirmed by computer simulations.152,154,156,166,168,193–196

Heterogeneity here is defined as some dihedrals exhibiting much fastermotion than average dynamics, whereas other (chemically equivalent) dihedralsexhibit much slower motion than average. The mean conformational transitiontime, which follows Arrhenius temperature dependence, is sensitive to fastevents. For example, a few fast dihedrals may exist in a system, with the remain-ing being quiescent. This system can yield the same mean conformational tran-sition time as a system in which all dihedrals have average dynamics. On theother hand, for the torsional autocorrelation function to decay, and hencefor 13C-NMR relaxation and dielectric relaxation to occur, each dihedralin the polymer must visit its available conformational states with ensembleaverage probability.

Heterogeneity in conformational dynamics can be quantified by measur-ing the distribution of waiting times for a given number of transitions. For PB,as shown in Figure 24,156 one observes an increase in probability for very shortwaiting times, which is from correlated transitions. Furthermore, a long timetail develops, indicating the existence of very slow dihedrals. These results arecompared in the figure with the expected Poisson behavior of independent ran-dom events. At high temperatures, the distribution approaches the expectedPoisson behavior. When reducing the temperature, the distribution becomesincreasingly heterogeneous.

Figure 24 Probability distributions for the waiting time for 10 dihedral transitions.Time is given in units of the average waiting time 10t . The distributions are peakedaround 10t ¼ 1 and are much broader than the Poisson distribution but approach it forhigh T. For low T, a high probability for short waiting times exists and a long time tail ofthe distribution develops.

Dynamic Heterogeneity 53

The dispersion of this waiting time distribution, i.e., its second centralmoment, is a measure that we can use to define a ‘‘homogenization’’ time scaleon which the dispersion is equal to that of a homogeneous (Poisson) system ona time scale given by the torsional autocorrelation time. The homogenizationtime scale shows a clear non-Arrhenius temperature dependence and is com-parable with the time scale for dielectric relaxation at low temperatures.156

The source of emerging heterogeneity (slow movers and fast movers) inthe conformational dynamics of amorphous polymers when decreasing thesystem’s temperature remains unknown. In light of the packing argumentsunderlying MCT, it is reasonable to attempt to associate differences in transi-tion rates with the local packing environment, i.e., a dense packing environ-ment for slow dihedrals and a looser packing environment for fasterdihedrals. The work by Jin and Boyd153 in this direction and efforts by dePablo’s group197 trying to relate heterogeneous dynamics to inhomogeneousstress distributions have so far been inconclusive. Phenomenologically, it isclear from simulations that conformational transitions become increasinglyself-correlated with decreasing temperature. (Here, correlation is defined interms of the probability that, once a dihedral undergoes a transition, this dihe-dral will then undergo another transition (usually back to the original state)before a neighboring torsion undergoes a transition.) It has been observedfor several polymer melts193,194 that the probability of self-correlationincreases dramatically with decreasing temperature. Self-correlation mayaccount for the ineffectiveness of conformational transitions to induce relaxa-tion with decreasing temperature: A relatively few torsions jumping back-and-forth rapidly between two conformational states can contribute significantly tothe rate of conformational transitions, but they will contribute little to theexperimentally observable local polymer relaxations.

The homogenization (or equilibration) process of the torsional transi-tions can be examined in even more detail. For example, one can differentiatebetween a first regime where every dihedral becomes mobile and is able to visitother isomeric states, and a second regime where the frequency of these visitsapproaches its thermal equilibrium value. Bedrov and Smith198,199 have ana-lyzed the time scales for these two regimes recently using modified PB-modelsand have shown that the first regime corresponds to the time scale of thedielectric b relaxation, whereas the second regime follows the a-relaxationtime scale. This study is a first inroad into a mechanistic understanding ofdielectric processes based on MD simulations.

SUMMARY

This chapter has the title ‘‘Determining the Glass Transition in PolymerMelts,’’ but we might ask: ‘‘Which glass transition?’’ Do we consider


the Vogel–Fulcher temperature T0, the calorimetric (or viscosimetric) glasstransition temperature Tg, or the mode-coupling critical temperature Tc tomark the transition?

The calorimetric or viscosimetric definitions of the glass transition tem-perature are arbitrary because they single out a temperature where the intrinsicrelaxation time of the glass forming system is approximately 100 s. When oneperforms either a calorimetric experiment or a volumetric experiment with sui-table cooling rates, a smeared step in the corresponding thermodynamicresponse function (specific heat or isobaric expansion coefficient) is observedaround Tg. When one is more patient in performing the experiment, as exem-plified by the classic work by Kovacs, the temperature at which the step in theresponse function is observed shifts to lower temperatures. For the class of fra-gile glass formers, to which most polymers belong, this shift is small, whichgives some validity to the arbitrary definition of Tg. For strong glass formers,however, the shift can be of the same order as Tg itself (for example, for silicamelts). Also, there is no qualitative change in any system-specific time scalenear the viscosimetric Tg. From a basic physical point of view, the calorimetricor viscosimetric glass transition temperature is therefore an inadequate mea-sure to use for defining the glass transition.

We have shown in the section on the thermodynamics of the glass tran-sition that an extrapolation of Tgð�Þ, the cooling-rate dependent step tem-peratures in the thermal expansion coefficient, postulating the empiricalVogel–Fulcher law as a description of the temperature dependence of internalrelaxation times extrapolates to Tð�! 0Þ ¼ T0 ¼ 0:35 for the bead-springmodel. This is in good agreement with what one obtains for T0 from the tem-perature dependence of the chain center-of-mass self-diffusion coefficient forthis model. Additionally, one generally finds that the so-called Kauzmanntemperature (where the extrapolated excess entropy of the super-cooledliquid in comparison with the crystal seems to vanish) agrees closely withT0. This agreement has led people to speculate about an underlying phasetransition around the Kauzmann temperature, especially based on theGibbs–Di Marzio theory for the excess configurational entropy of polymermelts, which also produced a Kauzmann paradox. We have shown, however,that, although the theory nicely reproduced the shape of the entropy curve asa function of temperature, the predicted absolute values were too small com-pared with the simulation data and the entropy catastrophe was just a conse-quence of inaccurate approximations. Furthermore, careful studies revealedthat the Vogel–Fulcher law cannot describe the temperature dependence ofthe a-relaxation time scale close to Tg. Instead, this temperature dependencecan be described by a low-temperature Arrhenius law to which the high-temperature dependence crosses over. Thus, the Vogel–Fulcher temperatureand the Kauzmann temperature are purely extrapolation artifacts having nounderlying physical significance.

Summary 55

The only remaining candidate for the definition of the glass transitiontemperature is therefore the crossover temperature from the high-temperaturebehavior to the low-temperature behavior of the a-relaxation time scale.A crossover that occurs around some temperature Tx that one also finds tobe the merging temperature of dielectric a- and b-relaxation and the criticaltemperature of mode-coupling theory. Physically, at Tx, a crossover occursfrom a high-temperature transport (or relaxation) mechanism to a low-tem-perature mechanism. For simple liquids, MCT identifies this crossover as ori-ginating from the cage effect. At high temperatures, the cage of neighboringparticles opens up on the same time scale it takes the central particle to reachits boundary. Contrarily, at lower temperatures, the cage particles are them-selves caged and an activated process is needed for the central particle to beable to leave its cage. This simple picture, and the theoretical predictions forthe behavior of the density fluctuations as being the dominating slow variableswithin the theory, have been tested on diverse systems like hard-sphere col-loids,200 Lennard–Jones mixtures,137 and silica melts.201 We have shownthat MCT can also describe fairly well the glass transition in a bead-springpolymer melt and in a chemically realistic model of PB. Deviations betweenMCT predictions and simulation results from chain connectivity, whichwere found in the bead-spring chain simulations, have been incorporatedinto an extension of the theory. For a chemically realistic model, we haveshown that a competition exists between two mechanisms, which leads to atime scale separation between vibrational motion and structural relaxation.One mechanism is related to the packing effects captured by MCT, and theother mechanism is from the presence of dihedral barriers. How these twomechanisms can be joined theoretically remains an open question.

The answer to our question at the beginning of this summary thereforehas to be as follows. When you want to locate the glass transition of a polymermelt, find the temperature at which a change in dynamics occurs. You will beable to observe a developing time-scale separation between short-time, vibra-tional dynamics and structural relaxation in the vicinity of this temperature.Below this crossover temperature, one will find that the temperature depen-dence of relaxation times assumes an Arrhenius law. Whether MCT is the finalanswer to describe this process in complex liquids like polymers may be apoint of debate, but this crossover temperature is the temperature at whichthe glass transition occurs.

ACKNOWLEDGMENTS

I am grateful to all my collaborators on the different aspects of the polymer glass transi-tion, and in particular I would like to thank J. Baschnagel, D. Bedrov, K. Binder, and G. D.Smith for a longstanding, stimulating, and fruitful collaboration. Funding through the GermanScience Foundation under Grant PA 437/3 and the BMBF under Grant 03N6015 is gratefullyacknowledged.


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Determining the Glass Transition in Polymer Melts ......transition in general.2,3 Instead of...

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