Dynamics of Glassy Polymer Melts in Confined Geometry: A Monte Carlo Simulation

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Geometry: A Monte Carlo Simulation

Jörg Baschnagel, Kurt Binder

To cite this version:

Jörg Baschnagel, Kurt Binder. Dynamics of Glassy Polymer Melts in Confined Geometry: A Monte Carlo Simulation. Journal de Physique I, EDP Sciences, 1996, 6 (10), pp.1271-1294.

�10.1051/jp1:1996137�. �jpa-00247245�

DynaTnics of Glassy PolyTner Melts in Confined GeoTnetry:

A Monte Carlo Simulation

Jôrg Baschnagel (*)

and Kurt Binder

Institut für Physik, Johannes~Gutenberg Universitàt, Staudinger Weg 7, 55099 Mainz, Germany

(Received

15 January1996, revised 4 June 1996, accepted 2î June

1996)

PACS.61.20.Ja Computer simulation of Iiquid structure

PACS.61.25.Hq Macromo1ecuIar and polymer solutions; polymer melts; swelling PACS.64.70.Pf Glass transitions

Abstract. Dynamic properties of _a dense polymer melt confined between two hard watts

are investigated _{over a} wide range of temperatures by dynamic Monte Carlo simulation. The temperature interval ranges from the ordinary Iiquid to the strongly supercooled melt. The _in- fluence of temperature, density aud confinement _on the polymer dynamics _is studied by _vanous

mean-square displacements, structural relaxation functions and quantities derived from them

(relaxation

limes, apparent diffusion coefficients, _monomer relaxation

rates),

yielding the fol- Iowmg results: The motion of the _monomers and polymers close to the waI1s is enhanced in

parallel, but reduced in perpendicular direction. This dynamic anisotropy strongly _mcreases during supercooling and extends into the bulklike _mner region of the film _over

a Iength scale which is Iarger than the bulk radius of gyration at Iow temperatures. However, the absolute freezmg of the melt

occurs in each Iayer at the _same temperature for both the parallel and the

perpendicular direction.

1. Introduction

The behavior of

polymer

^chains at an interface with _a solid represents an important

problem

in materials

design

because these interfaces

naturally

_appear in many modem technical _ap-

plications,

such _{as m}

Iubricants, polymer coatings

or liber

polymer-matrix

composites

Il,

2].

An

improved understanding

of these materials is

predicated

_upon

gaining

_a

deeper insight

in the characteristics of the so-called

interpha8e,

1.e., of trie interfacial

layer adjacent

to trie solid substrate.

Depending

_on the

specific

interactions with the substrate the interfacial

properties

of the

polymers

_can

considerably

deviate from the bulklike behavior

expected

outside of the

interphase [3,4].

If the interaction between the solid and the

polymers

is attractive, the

polymers

will adsorb at the substrate below the

adsorption

temperature. The adsorbed chains exhibit

substantially flattened,

almost two-dimensional

configurations [5-11],

whose structural relaxation

parallel

and

perpendicular

to the solid is

strongly

slowed down

[5-10,12,13]

and, _{m some} respect, similar to that of

glass forming Iiqmds [7,14,15].

This adsorbed

Iayer

forms _a

rugged

^surface

which limits the orientationai

freedom

and

mobility

of other

nearby polymers.

At meltlike deiisities trie

packing

constraints tend to orient

adjacent polymers parallel

to this surface.

(*) Author for correspondence. Present address: Institut Charles Sadron, 6 _rue Boussingault, 67083 Strasbourg Cedex. France.

je-mail:

baschnag©phebus.u-strasbg.fr).

© Les Éditions _de Physique 1996

This leads to characteristic oscillations of _{monomer or}

polymer density profiles. Typically,

trie

largest length

scale for trie

decay

of these oscillations is trie bulk radius of

gyratiou

R(~~~ at

temperatures far above the

glass

transition of the melt

là,

6,

9-12].

Since the adsorbed

layer

acts like _a

strongly repulsive

obstacle _to thé

adjacent polymers,

one can expect ^that a neutral solid substrate, where _no

preferential

attraction for _monomers

occurs, induces _a similar behavior. In

fact,

the mentioned

density

oscillations _are observed

m many off-lattice

là,

6,

9,10,16-23]

and lattice simulations

[24-28]

^of

polymer

melts close to

impenetrable watts,

and _can also be derived

analytically [21-23, 29-33].

This influence of confinement _on trie

polymer

structure also carries _over to the

dynamic

properties ^of the melt.

Computer

simulation studies indicate that the

mobility

of chains close to _a hard watt increases _m

parallel,

but decreases in

perpendicular

direction, relative to the

isotropic

bulk value

là,

6,

9,10,12,16].

This

anisotropy

is

usually rationjlized

_{as a} consequence of those chain

portions

which _are in immediate contact with the watt. Their

mobility

should be facilitated in

parallel

direction due to both the low

density

and the

preferentially parallel alignment

of the chains. In agreement ^with this

interpretation

_one finds that the motion of thé

polymers approaches

trie

isotropic

bulk behavior _on the _same

length

scale _as the chain

density profile,

_{1-e-, on} the scale of R(~~~

[5,6, 9,10,16].

The

dynamic anisotropy

observed in these simulations at

high

temperatures suggests ^that confinement

might

also have _a

pronounced

influence _on the

glass

transition of

supercooled polymer

melts. Which

changes

relative to the bulk behavior _are to be

expected

_can be inferred

from _various

experimental

studios of

glass forming polymers

in confined

geometry. Ellipsomet-

ric

experiments

^of thin

polystyrene films, spin-cast

onto _a

hydrogen-passivated

silicon

wafer, yield

_a decrea8e of the

glass

transition temperature Tg ^with

decreasiug

film thickness. This result is

interpreted

_{as a} consequence of _a

liquidlike layer

which is

supposed

to form at the free

(polymer-air)

surface and to

diverge critically

at trie bulk value of

Tg,

^{whereas trie} interaction with trie wafer is assumed to have _no substantial influence [34,

35]. Contrary

to

that,

_a recent

X-ray reflectivity study

_on

exactly

thé _same system reports an increa8e of Tg ^with

decreasing

film thickness. A decrea8e of Tg ^is

only

found when

using

_a silicon native-oxide surface [36]. ^In order to reconcile these

contradicting findings

^the authors of reference [36] point ^{out that their}

experiments were done in vacuum, while trie

ellipsometry

measurements of references [34, 35]

were

performed

in _an air

atmosphere. Exposed

to air trie

originally hydrogen-terminated

sili-

con surface is

likely

to be

quickly

oxidized and thus to exhibit _a behavior similar to that of _a native-oxide wafer. A silicon native-oxide surface is beheved to attract

polystyrene weakly

in

contrast to the

hydrogenated surface,

since _an increase of Tg ^with

decreasing

^{film thickness is}

found for other

polymers

for which thé interaction is known to be

strongly

attractive

[37-39].

This comparison of the various

experimental

studies

emphasizes

the

importance

of the precise

kuowledge

of thé

polymer-surface

interaction for _a correct classification of trie results. Since

experimentally

_one almost

always

bas to work with two

inequivalent surfaces,

which _{can even} act in diiferent

directions,

_a

complete understanding

of the data _may be _very diflicult. Due to this

difliculty

_a

freely-standing polystyrene

film _was used in _a recent Brillouin scattering

experiment

_[40]. Such _a film _{possesses two}

equiuaient (free)

surfaces _so that trie above described

problems

_are removed. This experiment shows that Tg ^{decreases with film thickness} and that

the extent of this decrease _is much

larger

than that observed in references

[34, 35].

In view of this stratified

phenomenology

further information for systems with known

polymer-

surface interaction is

certainly

beneficial to

complement

observations from

experiments.

The present ^{work intends} to contribute to this

by simulating

_a

simplified

model of _a

glassy polymer

melt confined between two

precisely equivalent, purely repulsive

watts in _a temperature interval encompassing ^the

ordiuary hquid

and the

strongly supercooled

state close to the

glass

transi- tion of the bulk.

Despite

its

simplified

nature the unconfined _version of the model

reproduces

many bulk

properties

of

experimental glass formers,

such _{as a}

strongly

stretched structural _re-

laxation,

_a non-Arrheniuslike temperature

dependence

of the

corresponding

relaxation times,

etc. [41]. ^{In addition to}

that,

the present ^model

yields

the

qualitative

structural features of confined

polymer

^melts at

high

temperatures, ^such as monomer

density oscillations,

enrich- ment of chain ends and

preferentially parallel alignment

of

polymers

at the

watts,

etc.

[42, 43].

Whereas the

Iargest Iength

scale for the

decay

of the watts' influence _on the melt structure is the bulk radius of

gyration

at

high

temperatures, the

corresponding

scale in the

supercooled

state turns ont to be

Iarger

than R(~~~ ^{and could be identified with}

that,

_on which trie _monomer

density profile approaches

the bulk value. This

finding

_means that the

interphase

^of a confined

polymer

melts

ezpands further during supercooling

than

expected

from its

high

temperature

behavior. That such _a macromolecular

interphase

could exist in the

supercooled

state close to the bulk

glass

transition _was also

conjectured

in the above mentioned

X-ray reflectivity [36,

39]

and Brillouin studies [40]. ^Therefore an

analysis

of the model's

dynamic properties

in the

light

of the obtained static results

[42,

43]

might help

to interprete these

experiments.

The present _paper is structured _as follows: The model and _some aspects ^{of the} data anal-

ysis

_are outlined in Section 2. Section 3 discusses the results of the

glassy polymer dynamics

in confined geometry,

paying particular

attention to the temperature

dependence

and to the amsotropy ^{of the} monomer

mobility

in

layers parallel

to the watts. The final Section 4 _sum- marizes _our conclusions.

2.

Methodology:

Simulation and

Analysis

In the present paper the

dynamics

of _a confined

(supercooled) polymer

melt is

investigated by

Monte Carlo simulation of the bond-fluctuation model [44]. ^{This model} represents ^the

polymers

as self- and

mutually avoiding

walks _{on a}

simple

cubic lattice. The

polymers

_are

built up ^ont of _monomers which _{occupy a} whole unit cell of the lattice. This

enlarged

_monomer

size

(relative

to

simpler

lattice models

[45, 46j)

entails _a multitude of _a priori

possible

bond vectors, ^{which is} reduced _{to a set} of108 allowed bond vectors

by

two conditions: local self- avoidance of the _monomers and

uncrossability

of the bond vectors

during

the simulation. The

resulting

set of bond vectors,

((2,

0,

0), (2,1, o), (2,1,1), (2, 2,1), (3,

0,

0), (3,1, 0)),

is

generated by

^ail symmetry

operations

of the

simple

cubic lattice.

Though finite,

thé available uumber of bond vectors sufiices to

approximate

the continuons space behavior

fairly

well _on

length

scales

larger

than _a few lattice constants

[47, 48].

^{On smaller} scales the

underlying

lattice

structure bas to appear. This _was quite apparent ^from our

previous analysis

of varions _monomer

density

profiles [42, 43]. However, although

thé

quantitative

features of these

profiles,

such as

the

regular zigzag

structure of the

density

oscillations

(at high temperatures),

the

amplitude

of these

oscillations,

etc., ^{is alfected}

by

the

lattice,

the

qualitative properties,

such _as trie enrichment of _monomers and chain ends at the

watts,

the

decay

of these

profiles

_on the

length

scale of _a

bond,

etc.,

correspond

to the results of

comparable

off-lattice simulations

[42, 43].

Since the

interphase

extends _over several lattice constants

already

at

high

temperatures and

seems to

expand during supercooling [42,43],

_we expect _our model to aise

provide qualitatively

reliable information _on trie interfacial

dynamics

of

(supercooled) polymer

melts.

In addition to chain

connectivity

and excluded volume interaction _an energy function

li(b)

is introduced, which favors bonds b of

length

b ₌ 3 and direction

along

^{the lattice} axes

(1.e., 7i(b)

=

0)

in

comparison

to the

remaining

bond vectors

(1.e., li(b)

= e)

[41,48].

The elfect of this energy function leads to _an expansion of the bond _vectors at low temperatures. ^A vector in the

ground

state

ii.

_e., in _one of the states of

((3,

0,

0)))

blocks four lattice sites which _may

no

longer

be

occupied by

other _monomers. Since the reduction of available volume increases with

decreasmg

temperature, _a

competition

between trie

energetically

driven expansion ^of

a bond and the local

packing

constraints in the melt

develops.

The

development

of this

competition

makes the structural relaxation time of the melt increase and induces the

glasslike freezing,

_{as numerous}

comparisons

of static and

dynamic properties

of trie unconfined mortel

with

experimental

and theoretical results show

[41,49].

This model _is extended

by

trie introduction of two hard walls to examine trie influence of confinement _on trie

glassy polymer dynamics.

Trie hard watts _are mserted in trie z-direction

at _{z =} 0 and _z

= H

= 40, whereas

periodic boundary

conditions _are used in trie other _two directions. Trie size of the simulation box in these directions is much

larger

than R(~~~ ^at ail studied temperatures

ii-e-,

L~

=

Ly

=

40)

to avoid any artificial interaction of _a chain with its

periodic

_images. The simulation box contains P

= 390

monodisperse polymers

of

length

N ₌ 10. Since each _monomer

occupies

8 lattice sites _on trie

simple

cubic

lattice,

the

density (volume fraction)

of the melt is # ₌

BNP/(H -1)L~

= o-à- This value is _an

adequate

_compromise between two

opposing

conditions: _{it is} low

enough

to allow for _a sufiicient acceptance rate of _{monomer moves}

(about

10% _at infinite

temperature)

and

high enough

to bestow trie

typical

behavior of dense

polymer

melts _on trie mortel [Soi.

In trie _course of trie Monte Carlo simulation _{a monomer} and _a lattice direction _are

randomly

chosen,

and _{a move is}

attempted

in this direction. If trie attempt does not violate trie excluded volume condition, trie _move is

accepted

with

probability exp[-AlilkBT],

where Ali is the energy dilference between trie

newly generated

and the

original

bond _vectors connected to the

moving

_monomer.

By

_means of this

probability

temperature is introduced in trie

simulation,

which _was varied from T

= I.o

(ordinary liquidlike state)

to T

= o.2

(strougly supercooled state)

_[51].

Since trie described _{monomer rnoves can be}

thought

of _as

stemming

^from a random force exerted _{on a monomer}

by

its environment, ^trie bond-fluctuation mortel exhibits Rouse-like

dynamics [41,52].

Therefore _we will refer _to trie Rouse mortel [53] ^to

interprete

and to

analyze

the obtained

dynamic

results. In this mortel the relaxation rate

W(T)

of _{a monomer is} related to the chain's diffusion coefficient

DIT) by

DIT)

=

~~~~~

(1)

Since the diffusion coefficient of _a

supercooled glass

former

depends

_on temperature in _{a non-} Arrhemus fashion which may be described, for instance,

by

_a Bàssler

equation [54,

55]

D(T)

=

DB

_exp

- )j ₍₂₎

or

by

_a

Vogel-Fulcher equation [54,

_56]

D(T)

₌ DVF _exp

(-)~

,

To

(3)

equation il) predicts

the _same temperature

dependence

for

W(T) (compared

with

this,

the weak temperature

dependence

of b~ [41] can be

neglected).

Therefore the

equilibration

of

a

supercooled polymer

melt is

extremely

difiicult

by

the local Rouselike

dynamics.

From _a computational point ^of view it is much _more ellicient _to work with

global

_moves which involve

a collective motion of ail _monomers of a airain. Such _a collective motion may be realized

by

trie so-called

siithering-snake dynamics

[45,

46],

^for instance. For trie present mortel this artificial

dynamics equilibrates

trie

polymer configurations

orders of

magnitude

faster than trie realistic Rouse

dynamics

_[42,

57].

Therefore _we used trie

slithering-snake dynamics

to generate

equilibrated

start

configurations

in trie studied temperature interval o.2 < T < I.o [51].

In order to reduce trie statistical uncertainties ten simulation boxes _were simulated in

parallel.

This _means that trie simulation involves 39000 _monomers. Whereas this _is sullicient _to

study

trie

glassy dynamics averaged

_over trie whole

film, layer-resolved properties

need _a

larger

effort to be determined with

adequate

_accuracy. To achieve this, _we

employ

trie

following

three steps

m

analogy

to trie method described in reference [12]:

1. deteruiine trie

z-position

of trie _monomers and of trie chains' center of _mass at t

= o

2. calculate trie

dynamic

properties ^with respect to these initial values _{over a} certain time

period (loooo

MCS'S for o.3 < T < I.o, 30000 MCS'S _at T

= o.25 and 70000 MCS'S for o.2 < T <

o.24)

3.

update

initial

positions

after this time

period

aud repeat trie

analysis.

Thèse

repetitions (typically

loo

-150)

_are used for statistical

averaging.

3.

Dynamic Properties

of the

Polymer

Film

In order to

study

trie influence of temperature,

density

and confinement _on trie

dynamic

_prop-

erties of trie

polymer melt,

various relaxation functions and uiean-square

displacements

_were

calculated _as averages over trie whole film and _as functions of individual

layers

situated at posi- tion _z above trie wall. From these

dynamic

quantities trie temperature and trie

z-dependence

of structural relaxation times and transport coefficients _can be extracted and

compared

to trie _un- constrained bulk behavior of trie

supercooled

uielt. Trie results of this

analysis

_are summarized in trie

following

sections.

3.1. ACCEPTANCE RATE. One of trie

simplest dynamic quantities

to calculate is trie accep-

tance rate

A(z)

which _measures trie number of successful _monomer

jumps. A(z)

is defined

by

~

number of

accepted

_monomer

jumps

at _z

~~~ ~

total number of _monomers at _z ^'

where _{a monomer is} associated with trie value _z if trie lower face of its unit cell

belongs

to trie zth

layer

of the cubic lattice. To determine

A(z)

it is not necessary ^to monitor trie time evolution of trie

melt,

but it sullices to test for trie movableness of _monomers in

snapshot configurations.

Therefore all results to be

presented

in this section _were obtained

by

thé

slithering-snake algorithm (statistics

based _on 20000

independent configurations,

1.e., _on 7.8 _x 10~

monomers

in

total).

Figures

la and 16 show trie

profiles

of the

perpendicular

and

parallel

_components of trie acceptance rate at T

= I.o and T

= o.18. The

parallel

component Ajj ^{is the} average of trie acceptance rates in trie +z- and

+y-direction.

This

averaging

is

legitimate,

since trie acceptance rates in trie varions

parallel

directions _agree with each other within thé _error bars at both temperatures. This _means that thé motion

parallel

to wall _romains

spatially

isotropic

m mdividual

layers

not

only

at

high

temperatures, but also in trie

strongly supercooled

state.

Since trie

profiles

_are

symmetric

around the middle of the

film,

the characteristic properties will be discussed for trie left

half,

_{i e.,} for o < _z < 20.

At T

= 1.o Ajj ^is

slightly

enhanced above trie bulk value Abuik

Ii.

_e., the average of

A(z)

for 15 < _z < 25 [42]

adjacent

to the wall

(z

=

1),

whereas it is _{more or} less bulklike at T

= o.18.

At this low temperature a

(strong)

enhancement _occurs in the second

layer

above trie watt at

z = 2. This dilference _can be rationalized _as follows: At

high

temperatures ^trie structure of the

polymers

^close to the watt _is determined

by

the

competition

of two opposing forces: the force

resulting

from the loss _m

configurational

entropy due to the presence of the

impenetrable

0.35

0.3

(( Î~

parallel

-

0.25 0.155

d 0.2

« 0.15

o-i

o.05

o

0 5 10 15 20 25 30 35 40

a)

^z

o.06

0.05 "Z ~

jaral~l

^-

O.M ,00674

ç

^0.03

<

0.02

O.oi

o

0 5 la 15 20 25 30 35 40

b)

^z

Fig. l. Distance profile of the acceptance rate

A(z)

for _monomer jumps perpendicular

((Q):

_-z;

ix ):

+z)

and parallel

(Ô)

to the waII at T ₌ I.o

(a)

and at T

= 0.18 16). The horizontal solid Iine mdicates the bulk value (Abuik

" 0.155 at T

= 1-o and Abuik

# 6.74 _x 10~~ _{at T}

= 0.2) which _was determined _as the average of A(z) for 15 < _z < 25.

wall

(depletion force)

and that

resulting

from collisions with other further distant

chains,

which tends to

pack

the

polymers against

the wall

(enrichment force).

The dominance of the

packing

^force at meltlike densities enhances the _monomer and trie end _monomer concentration at trie

watt,

which in tum leads to _a reduction of the respective concentrations in the second

layer

because of the excluded volume interaction [42]. ^{Since trie} movableness of chain ends is increased in

general,

and trie parallel motion of inner _monomers should also

fairly easily

be

possible

due to both trie free

ghde along

trie wall and trie

depletion

of trie second

layer,

_one

can understand that

Ajj(z

₌ 1) is

larger

than trie bulk value.

With trie _same

reasoning

it is

possible

to

explain

trie

spatial

variation of Ajj. ^At z = 2 trie chain end

density

is

small,

whereas the overall _monomer concentration _m the

adjacent layers

is

high.

This should entait _a small value of

Ajj.

^At z = 3 the situation is _agam

comparable

to that at _{z =} 1, whence _a

large parallel

acceptance rate is to be

expected.

Therefore the

spatial

variation of

Ajj(z)

^{cali be} rationalized _{as a} consequence of the

oscillatory

structure

of trie _monomer and end _monomer

profiles.

However, the oscillations of

Ajj(z)

are far less

pronounced

than those of the _monomer

profiles.

This _means that trie wall-induced

spatial inhomogeneities

influence the

parallel

acceptance rate

only weakly

at

high

temperatures.

This situation does not

change

with

decreasing

temperature,

although

thé

length

scale for trie

decay

of the _monomer

profiles

increases

strongly

_[42]

(sec Fig. 1b).

At low temperatures trie model's _energy function becomes

progressively important

for trie

polymeric

structure in the

interphase.

Its

interplay

with

density

and watt elfects tends to

align

successive bonds in trie

ground

state

parallel

to the wall [42]. ^This

considerably

enhances the number of _monomers,

especially

trie number of inner _monomers, at the wall. Therefore _{a monomer}

jump

is

likely

to increase trie _energy, which will be

exponentially suppressed by

trie Boltzmann factor.

Contrary

to

that,

trie enrichment of chain ends in trie first and trie

depletiou

^{of trie} second

layer

favors trie

parallel

motion. Since bath eifects work in opposite directions, trie value of

Ajj(z

= 1)

relative to trie bulk at T ₌ o.18 should be reduced

compared

to that _at T ₌ I.o. That

Ajj(z

=

1)

^{almost coincides with trie bulk value is}

certainly

accidental. At _{z =} 2 trie number of bonds in trie

ground

state is

small,

whereas it is

large again

at z = 3 [42]. ^As

Figure

16 shows, this

spatial

variation of trie number of

ground

state bonds _seems to dominate _over trie

correspondiug antiphasic

behavior of trie _monomer

profiles

for _z < 3, but to be

compensated by

it for

larger

z-values- This

compensation presumably overdamps

trie oscillations of

Ajj(z)

for _z > 3 and makes trie

profile approach

trie bulk value faster than at T

= I.o.

Although

this

rapid approach

is

certainly

a consequence of trie studied model, trie

Ajj(z)-profile

^should

decay

_{on a} small

length

because trie

parallel

motion is

(to

_a

large extent)

determined

by

trie

monomer

density

in trie

loyers

_z -1, z and _z +1 and thus _{averages over} them. This _average

monomer

profile

reaches trie bulk value faster thon trie actual _one.

Contrary

to that, trie

perpendicular

acceptance rate towards A- and _away from trie watt

A+

is controlled

by

^trie monomer

density

in

layer

_z 2 and _z + 2,

respectively.

Since

jumps

_away from trie watt

point

^{outside of} trie

interphase

into trie bulklike

region

of trie

film,

whereas those towards trie watt _occur

against

^trie

strongly oscillating

_monomer

profile,

_{one con} expect

A+

to reach trie bulk value earlier than A-. This

expectation

is borne ont in

Figures

la and 16. At both temperatures trie

impenetrable

watt prevents

adjacent

_monomers from

jumping

towards it. Therefore

A-(z

=

1)

₌ o. Trie

corresponding

rate for

jumps

away from trie watt is much

larger

than

A-(z

=

1) (and larger

than

Abuik)

at T

= 1.o, but almost

equal

to

A-(z

₌ 1) at T = o.18. This temperature

dependent

^diiference is _a result of the

large

number of bonds in the

ground

state at z = 1 and of the

high

_monomer

density

at _{z =} 3 [42]. ^{Both eifects inhibit}

jumps

_away from the wall. For _monomers at _{z =} 2 the situation is inverse.

Jumps

towards the watt _are

only

limited

by

chain

connectivity,

whereas those away from the watt

additionally experience

the

repulsive

_monomer

density

at _{z =} 4. Due to the low _monomer

density

at z = 4 and trie small number of

ground

state bonds at z = 2

(at

T

= o.18 [42) one can understand

why A-(z

=

2)

>

A+(z

=

2)

> Abuik at both temperatures. Trie _same kind of

reasoning

_may

be

applied

to rationalize trie

profiles

of trie

perpendicular

acceptance rate for further distant

layers.

^Since trie

A-(z)-

and trie

A+(z )-profile

therefore reflect trie _monomer

density

at _z 2

and _z + 2,

A-(z)

reaches the bulk value _{on a}

larger length

scale

(about

4

layers later)

than

A+(z).

An average

profile

for the

perpendicular

acceptance rate,

however, decays

_on the _sanie

length

scale _as trie _monomer

profile.

This

length

scale

corresponds

to that of _a bond at

high

temperatures, but is

larger

than trie bulk radius of

gyration

at low temperatures [42].

-2 ^.° o o

._ a

-2.5 o

_

-3

~

-3.5

<

~ ~

~.5

-5

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

1/T

Fig. 2. Arrhenius plot of trie bulk acceptance rate Abuik

Ii.

_e., the average of

A(z)

for là < _z < 25;

see Figs. la and 16). The solid and the dashed Iines _are fits to _an Arrhenius and _a Bàssler equation

(see

Eq.

(2))

for 0.18 < T < 0.25.

In order _to obtain _{a more}

quantitative insight

in trie temperature

dependence

of trie _accep- tance rate

Figure

2 presents an Arrhenius

plot

of trie bulk value Abuik

Ii.

_e., trie _{average over} trie inner part of trie film for 15 < _z < 25 [42]) ^and compares trie simulation data with _a fit to trie Arrhenius and trie Bàssler function

(see Eq. (2)).

For both functions trie fit interval is o.18 < T < 0.25. Trie

figure

shows that the Arrhemus fit is superior to the Bàssler fit in this low temperature interval close to trie

glassy freezing

^{of trie} mortel [41]. ^{A similar} observation _was also made _m another work with trie bond-fluctuation

model,

in which trie _energy function _was tailored in such _{a way} that

large length

and time scale properties of

bisphenol-A-polycarbonate

coula be simulated [58]. ^{Trie Arrhemus fit}

yields

^{for trie}

amplitude A(oc)

and activation energy EA:

A(oc)

= 1.65 + 0.03 and EA

= 0.989 + 0.010 _{m e.} This result is

typical

of _a two-level

(TL)

system, as present ^{in trie} simulation

by

trie model's _energy

function,

since

(A)TL

_"

P0A0-e

+

PeAe-0

~~~°

~°

~ÎÎÎ

^~

^~Î

=

Ae-oge~)~~ (('

«

expl-fiel

,

là)

TL

where _po

"

go/ZTL (with

_go

"

6)

and _{pe =} ge

exp[-fie]/ZTL (with

_ge

=

102)

are trie

probabil-

ities of

finding

_a bond in trie

ground

_or in trie excited state. In trie last fine of equation

(5)

detailed balance _was

applied

and trie temperature

dependence

of ZTL _was

neglected

due _to ge

exp[-fie]/go

_< 1 in trie used fit interval. This

simple

calculation suggests that trie low tem-

perature ^Arrhenius behavior of Abuik is

predominantly

caused

by

trie model's _energy function.

Density

eifects determme the value of the

prefactor,

^but seem to influence trie activation energy

only weakly.

In the _same way the temperature

dependence

of trie acceptance rate ofindividual

layers

_can be fitted

by

_an Arrhenius

equation. Figure

3 _summarizes trie result of these lits

by comparing

-z -

~'~

paral~l

_-

1.2

« 1

uJ

0.8

o.6

0.4

2 4 6 8 10 12 14

z

Fig. 3. Distance profile of trie Arrhemus activation energy EA for trie acceptance rate

A(z)

of

monomer jumps perpendicular

((Q);

_-z;

ix

):

+z)

and parallel

lé)

to trie waII. At ail z-values the

fit innervai for the Arrhenius equation _was 0.18 < T < 0.25.

trie

spatial

variation of EA for A-,

A+

and

Ajj. Contrary

to trie

bulk,

trie

density

_now bas

a

pronounced

influence _on trie value of EA, which may be rationalized

along

trie _same fines

as for trie acceptance rate

profiles.

Whereas trie activation _energy for

parallel jumps quickly approaches

the bulk value

ii-

_e.,

starting

from _{z m} 71 due to trie above mentioned

averaging

over

several

layers, EA

for A- and

A+

reflects the _monomer

density

at z-2 and z+2

(large

EA-value if

density

is

large

and vice

versa).

Therefore EA for the

perpendicular

components oscillates

much stronger than for trie

parallel

component, and its average over the two

perpendicular

directions becomes bulklike _{on a}

larger length

scale

(1.e., starting

from _{z c5}

12).

Although

the acceptance rate

already

reveals the asymmetry ^between

parallel

and

perpendic-

ular motion in the

interphase,

its temperature

dependence

does not

signal

the

glassy freezing

of trie melt. Trie _reason for this is that it _courts

oscillatory jumps

_as successful _monomer

moves,

although they

do not contribute to structural relaxation. Whereas these

oscillatory

jumps

can therefore

persist

in the frozen state down to T

=

0,

the structural relaxation of the melt could _cease at _a

higher

temperature.

Appropriate quantities

to

investigate

^{this behavior}

are structural relaxation times and transport coefficients.

They

will be discussed in trie _next sections.

3.2. PROPERTIES _AVERAGED ovER THE WHOLE FILM.

Figures

4a and 4b show trie time

and temperature

dependence

of two

representative examples

of

polymeric

relaxation

functions,

1.e., ^of trie

perpendicular

component ^{of trie bond vector} correlation function

~ ~~

ibi(ti bi10)1 ibi(ti) ibi(0)1

~~~~ ~ _~~~

~'~

ibi10)) ibi1011~

^{~ ~}

and of trie

parallel

component ^of trie end-to-end vector correlation function

~~'~~ ~~~

~~~~~~

IÎÎÎI ÎÎÎÎÎÎI~~~~~~~~

_~~~~

~'~ ~~~'~~~

_~~~

1 1~f

.,

.

~

++

^'++~~,

+~ ~

0.8 G~ °o.

~

~ o~

, , , +, _,

. a ~ a + ,

, ,

~

o +~ ~

-

06 » a x j

~ ~

- » a x ~ ^{+ o}

~ symbol $ § ~+ °o

w 02~ ~ ç j ~ a ⁺ ~

~ _0A

0.21 + ~

)

$

) ^Î

Î

o~ ~ ^{. x +}

o

. ^a x o

~ ~

0.25 _x j _xx $

++ °,

~'~

~~

^' ~~ ^~

~ ~

~

'& _~& S

o

0 1 2 3 4 5 6 7

a~

'°gwt

1 _{. »»}

~,

++~

~~

'++

~ '

WÀ'

, ^~ i~

q~S

^%

symboj

Ù

x

S $~

C ^{°.6 0,20 o}

~ i Î

^{~ ~}

)

021 ₊ ^A j $

+ ~

~

0 4 ~.Î~

Î _~~~

035 . ~ x

1.00

~

* x

$

~

» x +

" Î Î

~

~

~~

. ~ ~ +

' 4

~ x a

~ $ x %

' ~~_ ^~

0 ~

0 2 3 4 5 6 7

b) '°gw

t

Fig. 4. Plot of

4lb,1(t) (a)

and 4lR,jj(t)

(b)

_{versus Iogj~t} for different temperatures _as specified in trie figure.

In trie studied temperature interval trie relaxation of both correlation functions slows down

by

about 3 4 orders of

magnitude. Despite

this strong increase of trie

polymer

relaxation time thé correlation functions maintain tiroir

high temierature

_form.

_They

seem to be

self-8imiiar

in

trie _sense that _a decrease of temperature

merely

shifts trie functions

along

the abcissa to

larger

time values without

alfecting

their

shapes.

If this _was true, it should be

possible

to

collapse

the data at dilferent temperatures onto a temperature

mdependent

master curve

by rescaling

trie abcissa values with _a

smtably

defined, temperature

dependent scahng

time t~~. Since such _a lime-temperature

superposition

is

theoretically predicted [49,

54] ^and

experimentally

observed [49, 59, 60] ^{for fate}

times,

_a

possible

choice for _{t~~ is} the time value when 4~(t) m o-à-

Figures

Sa and 5b present _a test of this idea. Trie

scaling

does not

only

work _very well _at

large times,

^{but also} over the whole time range. This _means that there is _one time scaie which

1

f

0.8 0.20 _o

0.22 +

0.25 D

= ^0.6

(.~

)

l.00 _»

~

0.4

0.2

o

-6 -4 -2 0 2

~j '°Ilw t'tsc

1

~~ symbol

0.20 o

~~

0~22 ₊

C 0.25 _o

Î 0.35 x

ù~ 0.40 A

* _0A 1.00 »

0.2

o

-7 -6 -5 ~ -3 -2 -1 0 2

iog~ tit~~

b)

Fig. 5. Scaling plot of 4~b,1Ii)

(a)

and 4lR,jj

Ii) (b)

_versus

logj~

t

fisc

for different temperatures _as specified in the figure. tsc is taken to be tI1e time value when 4lR,jj(tsc) ₌ 4~b,1(tsc ₌ ^0.4749.

determines the whoie temperature

dependence

of the

respective

correlation functions.

The temperature

dependence

of the

scaling

times, obtained from

4~R,jj(t), 4~R,1(t),

_4~b,jj(t)

and

4lb,1(t),

_is

depicted

in

Figure

6. The

figure

shows that _{t~~ of} the end-to-end vector _corre- lation is

larger

than that of bond vector correlation in the studied temperature range. At

high

temperatures ^{the dilference} is

approximately

_an order of

magnitude (1.e.,

about _a factor of

N),

but becomes smaller with

decreasing

temperature

(1.e., by

about _a factor of

2). Contrary

to

that, the values of the

parallel

and

perpendicular scaling

time (t~~,jj and

t~~,i)

which

nearly

coincide at

high

temperatures,

gradually

separate ^with

decreasing

temperature. ^{Whereas the} dilference between trie bond and the end-to-end vector correlation function _may be

anticipated

(larger length

scales

along

_a

polymer

relax _on

longer

^time

scales),

trie smaller

t~~,i-value

at low

le+07

1e+06

o

_#

le+Ù5 o

Ie+04

Ie+03

0.2 0.25 0.3 0.35 0A

1

Fig. 6. Temperature dependence of trie scaling times tsc obtained from 4~R,jj

la

), 4~R~i

(O),

4~b~jj

(+)

and 4~b,1

lx

). TI1e dashed and the sohd Iines _are lits to equations (2) and (3) for tac from 4~R,jj and 4~b,1 tsc _is taken to be the time value when tI1e various correlation function _are equal to 0.4749.

temperatures is

unexpected

_on trie basis of trie results to be

presented

about trie monomer's and chain's

mobility (see

Sect.

3.3).

It _can be rationalized _as follows: The studied

polymeric

correlation functions _measure the _average

length

and reorientation fluctuations of the bond and end-to-end vectors. In trie

vicinity

^of the wall both bonds and chains _are

preferentially

oriented

parallel

to the

wall,

which entails _a small

perpendicular

component ^of the respec-

tive _vectors [42]. ^With

decreasing

temperature ^the wall-induced orientation of the

polymers

still reinforces. These

strongly aligned

chains form _a kind of _a

rugged

^surface m front of trie smooth

wall, against

which further distant

polymers

_are oriented. Therefore the

perturbation

introduced

by

trie hard watt penetrates

deeply

into the bulk and makes trie

interphase expand during supercooling

[42]. ^Trie

interphase

thus contributes

considerably

to the _{average proper-} ties of the film at low temperatures.

Although

trie

perpendicular mobility

_m trie

interphase

is smaller than trie

parallel (see

Sect.

3.3),

trie

perpendicular

motion _can

strongly

reduce trie

corresponding

vector components

(transition

from _a finite value for bz to zero is

possible)

_so

that _{on average}

t~c,i

< t~~,jj at low temperatures.

In addition to trie simulation results

Figure

6 shows lits to trie Bàssler

(Eq. (2))

and

Vogel-

Fulcher

equation (Eq. (3))

for trie

largest

and trie smallest

scahng

times

ii.

_e., for

4lR,jj(t)

^and

4lb,1(t)).

For trie

Vogel-Fulcher

fit trie interval o.2 < T < oA _was

used,

but for trie Bàssler fit it had to be restricted to o.2 < T < o.3 to obtain _a

comparable quality.

Trie outcome of this

analysis

is summarized in Table I. Due to trie

larger

interval trie

Vogel-Fulcher

fit is

slightly

better than that

by equation (2). Nevertheless,

_since both trie Bàssler and trie

Vogel-Fulcher equation

work

equally

well in trie interval o.2 < T < o.3, it is hard to

distinguish

between _an absolute

freezing

at finite _{or at zero} temperature

by

_means of trie present data.

However,

trie fits

suggest

^{that trie bond and end-to-end} vector components ^freeze at trie _same temperature.

Another

interesting quantity

in this _context is trie diffusion coefficient

parallel

to

watt,

which

is defined

by

Djj

= ii~

93,jj(t)

_g~~~j~)

~~~ ~~ ~~

t=t~~~(T~

~~~~ ~P,11 ^"

(Rp,~,R~

_~)