Static and dynamic properties of two-dimensional polymer melts

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Submitted on 1 Jan 1990

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Static and dynamic properties of two-dimensional

polymer melts

I. Carmesin, Kurt Kremer

To cite this version:

Static and

_{dynamic properties}

of two-dimensional

_polymer

melts

I. Carmesin

₍₁₎

and Kurt Kremer

₍₂₎

(1)Max

Planck-Institut für

_{Polymerforschung,}

D-6500

_Mainz,

F.R.G.

(2)Institut

für

_{FestkOperforschung,}

KFA

Jülich,

D-5170

_Jülich,

F.R.G.

(Reçu

le 8 novembre

_1989,

_accepté

le

_{7 février}

₁₉₉₀₎

Abstract. 2014 We

present a detailed

_{analysis of}

the

_properties

of dense linear two-dimensional

_polymer

chains. The

_systems

are simulated

_by

the

_{recently developed}

bond fluctuation method. We

_investigate

systems

of _up to 80 %

_density.

The chain

_length

and densities considered cover the crossover from the

_expanded

_single

chain limit to a dense

_polymer

melt. The chains

_completely

_segregate and follow

a _power law of the chain extension

R2

>~

N203BD,

N

_being

the number of bonds of the chains and 03BD =

1/2.

While for

single

_two dimensional chains the Rouse modes _{are not}

_{eigenmodes they}

are

_eignemodes

for the chains in a 2-d melt. Relaxation functions and mean _square

_{displacements}

display

typical

Rouse-like

_behavior,

with the

_distinction,

that the

_prefactors

for the various relaxation processes differ.

Classification

Physics

Abstracts 61.40-66.10c

1. Introduction.

Most

_{experimental investigations}

of

_polymers

consider chains in three dimensions. This is the

most natural

situation,

however,

there are _many exact theoretical results for d = 2

[1-3].

Thus,

a two-dimensional

_polymer

_system

would be an ideal

_testing

case _{for many} modern theoretical

concepts,

such as renormalization

_group

and conformal invariance methods. Besides this there is not much known about the

_dynamics

of such

_systems

_[4].

_Again

a two-dimensional

_system

should be ideal to

_investigate

the Rouse model

_{experimentally.}

For d = 3

hydrodynamic

interactions

always

dominate the behavior of thé chains in solution. For d =

2,

_e.g. chains near a

_surface,

this

probably

is not the case since

_long

_range

_hydrodynamic

_{interactions may} be

_completely

screened

out. For two dimensional melts the situation is

_especially

_interesting.

The

_topological

interaction is

_completely

different

_compared

to the d = 3 case.

_Following

recent theoretical

_{investigations}

the chains should

_collapse

and

_segregate

_[1].

However,

how does the

_shape

of the chains look like? Do

_they

_appear

as ideal random

_walks,

_although

the

_topological

_aspect

of the excluded volume interaction is a dominant

_aspect

for d = 2? Recent numerical simulations were not able to discuss this

_question

in detail

_[6-8].

Since

_they

use bond

_breaking

simulation methods their

samples

are,polydisperse.

As a

_consequence

there is a

_mixing

of

_long

and short chains

_leading

to

an effective

_expansion

of the

_longer

chains. The classical

_papers

of Wall et al.

_[9]

were not able

to address the above

_questions.

_Baumgârtner

_[10]

used the

_{reptation algorithm}

to

_analyse

the

916

differences between the ideal random walk structure and the 2-d melt chains. There the

_analysis

was confined to the ratio of the radius of

_gÿration

and the end to end distance.

_Bishop

et aL

_[11]

use a Brownian

_Dynamics

method in order to

analyse

2-d melts at rather low densities and for

short chains

_{( N}

_50,

p =

0.50).

In addition not much is known about the

dynamics

of dense 2-d

systems.

A recent

_{investigation}

of the

Rouse/Zimm

model of 2-d chains

_[12],

of course, could not

take into account the

_topological

_aspects

of the

_{problem. Experimentally}

these

_systems

are also

not

_completely

out of reach. Modern technics in

_producing

various

_polymers

could _e.g. construct

chains with small side branches

_preferring

two different solvents. If these solvents are

_immiscable,

such

_polymers

would be located at the interface between the two solvents. One can also think of chains between

_lipid

_monolayers.

Also recent

_developments

in the surface

_coating

of mica sheets

[13]

may

lead to

_stricly

2-d

_systems.

_Finally,

of _{course, the} classical

_approach

of

_spreading

chains out of a volatile

_good

solvent onto a

_liquid

air interface

_may

be used. After some

_{equilibration}

time one then can

_compress

the film towards a semidilute or dense

_layer.

Of _{course, there} one

ltas to take care not to

_get

double or multi

_layers.

Considering

these

_{developments,}

we think that it is

_very

desirable to have a better

under-standing

of such

_systems.

Thus we

_performed

an extensive numerical

_study

of a 2-d

_many-chain

system.

To do this we made use of a

_{recently developed}

Monte Carlo

_algorithm

_{[8, 15].}

This bond fluctuation method was used to

_analyse

the

_properties

of an isolated 2-d chain. The way the

moves are constructed also allows to

_study

the

_dynamics

of 2-d

_polymers

_[15].

We use this method

to

_study

_systems

with

_up

to 80 %

_density

and chain

_length

of

_up

to N = 100 monomers. A short

preliminary

account of this work is

_given

in reference

_[16].

Chapter

two

_gives

a short

_description

of the

_algorithm

and the

_systems

considered. Then we

describe in some detail the

_{équilibration}

_process.

_Chapter

three

_gives

our results for the

_statics,

while

_chapter

four

_gives

the

_{dynamics. Finally}

_chapter

five contains the conclusions and a short outlook.

2. Model and methods.

In order to

_investigate

the

_{dynamical properties}

of a 2-d

_polymer

_system

_by

mean of Monte

Carlo,

we have to assure that the

_{algorithm displays}

Rouse

_dynamics

for the non reversal random walk

_[8].

This random walk assures that consecutive monomers are not allowed to sit on

_top

of

each other. Besides this short _range

_{repulsion along}

the chains no additional interaction is taken into account. For such a

_system

one can use the

_complete

set of moves as for chains with full

ex-cluded volume. This then

_gives

a

_good

and

_dependable

check on the static and

_dynamic

_properties

of an

_algorithm.

For

_dynamic

Monte Carlo

_algorithms

the

_typical

_requirements

for the moves _are,

that

_they

create new bond vectors within the chain. Standard lattice methods for d = 2

only

cre-ate new bond vectors at the

ends,

which then diffuse into the chain. This enhances the relaxation times

_{artificially.}

To overcome this

_problem

the bond fluctuation method was introduced

_recently

[15].

It combines the

_advantage

of a lattice

simulation,

which means

_e.g.

fast

_algorithms

on both

scalar and vector

machines,

with continuum

_approaches,

so that new bond vectors are created in

successful moves.

_Figure

1 illustrates the method. Each monomer

_occupies

four lattice sites. The bond

_{length 1}

is restricted to be

₁

_13.

In

addition,

a lattice site can never be

_occupied

_by

more

than

one monomer. For a move, a monomer is selected at

_random,

then it

_jumps

at random

_by

one

lattice distance into one of the four lattice directions. If the new

_position

_complies

with

_both,

the bond

_length

restriction and the excluded volume

restriction,

the move is

_accepted

and otherwise

Fig.

1. - Illustration of the bond fluctuation method. Ibe

typical

moves are indicated.

Note that the initial state must not contain crossed bonds. Since the bond

_length

constraint is

set to

_prevent

_crossing

of bonds such a

_{configuration}

would be

_preserved

forever. For dense

sys-tems the first blocked

_{configuration,}

which is a situation the

_system

cannot reach or

_leave,

occurs

at

_{density exceeding}

80% for infinite chains. Since this

_{configuration}

is

_ordered,

its

_probability

is

_{vanishing compared}

to the other random coil states.

in

the

_{subsequent investigation}

we

_only

use densities of

_up

to 80%. Since our chains are finite

_(N

₁₀₀₎

there will be no

_problem

of unresolvable blocked

_{configurations. (There}

is a _very small

_probability

that a random initial

configuration

of a chain is blocked

_by

itself. These

_{configurations}

have a local

_density

of one.

The initial conditions

_employed

in the

_present

work excluded such

_{configurations automatically).}

The simulations were

_performed

on a 100 x 100

_square

lattice with

_periodic

_boundary

condi-tions and densities with

_{20%, 40%, 60%,}

80% of the lattice sites

_,occupied.

We

_analysed

chains with

₂₀

N 100. The N = 100 and 80%

density

_system

then constains 20 chains. The

calculations were

_performed

on a

_Fujitsu

VP

100wector

_processor.

_typically

100

_{statistically}

in-dependent

systems

of

thé

same kind were simulated in

_parallel

The vectorization was set

in

a

way that one monomer in each

_system

was moved

_{simultaneously. By}

this

_approach,

even

_during

the standard simulations

_part,

we were able to vectorize the

_program

_efficiently

_[18].

lb start our

systems,

we filled the lattices with ordered

_arrays

of chains of

_length

N = 100 _monomers and

80%

_density.

The chains all had a U-like form and were

_pairwise

nested into each other.

Fig-ure 2a

_gives

a

_snapshot

_picture

of such a

_system

_shortly

after the simulation started. One

_easily

identifies from this the initial

_type

of

_{configuration. Figure}

2b

_gives

a

_{typical snapshot picture}

of an almost

_equilibrated

structure of the same

_sample.

The three different marked chains

_give

the

_typical

cases. 1Bvo chains have a

_fairly

smooth round

surface

while the third is still

_strongly

stretched. The stretched one

_(which

is the extreme of that

_sample)

_displays

a rather unfavourable

configuration indicating

that the

_system

is not

_yet

in

_complete

_{equilibrium. During}

the actual

918

Fig.

2. -

_(a)

_{Snapshot configuration}

of a p = 0.8,

N = 100 _system a short time after the initial state. The internested structure is

_{clearly displayed.}

_(b)

_Snapshot

_{configuration}

of the structure of

_figure

_2a,

where the

system

was almost

_{equilibrated.}

Fig.

3. -

Typical equilibrated

structures for

_a)

p =

0.2,

N

_{= 100;}

_b)

p =

_0.4,

_N = 20.

has the same

_amplitudes

while in the very

beginning they

were _very

_asymmetric

due to the initial condition. This

_provided

a sensitive test of

_{equilibration.}

The short

chain/low

_density

_systems

then were made from the N =

100,

_p = 0.80

sample

simply

_by

_cutting

bonds and

_eliminating

-

had started the

_systems

from scratch

_again.

The

_special

initial condition was used to make sure

that no

_memory

was

_{kept during}

our simulation and to check the

_proposed

_segregation

_[1].

The initial internested structure is

_specific

_enough

to follow its

_decay

in détail. After

_{equilibrating}

the

_systems

we followed the motion of the chains at least

_up

to a distance of their own diameter.

Considering

that we

_always

ran 100

_systems

in

_parallel

this

_gives

even for

_{the p}

=

0.80,

N = 100

system

20 x 100

_independent

diffusion

_paths,

which is

_enough

to

_provide

data of

_good

_accuracy.

Figure

3

_gives

two other

_typical

_{configurations}

at

_density

_p = 0.20 and _p = 0.40.

3. Static

_properties.

In

_polymeric

melts the excluded volume

effect,

which causes the

_expansion

of chains in

_good

solvent

_{[2, 19, 20],}

is

_expected

to be

_effectively

screened out

_by

the interaction

_among

the

differ-ent

_polymers.

This is known

_{theoretically}

to hold also for d = 2

[1, 2].

The

major

différence

between d = 2 and d _>

2,

however,

is

_that,

due to the

_topological

constraints in d =

2,

the chains cannot

_entangle.

If

_they

want to

_{interpenetrate,}

this

_only

is

_{possible by partial alignment}

of the chains. This

_certainly

causes a reduction of

_entropy.

The

_consequence

_is,

that the chains are

_going

to

_segregate.

One of the

_{interesting questions}

now

_is,

how

_strong

is the

_{segregation depending}

on

_chainlength

and concentration. The

_equilibrium

structure of chain is

_supposed

to be like a

Hamiltonian walk.

In order to

_investigate

the structure of such a

_polymer

_melt,

we first check the mean

_squared

end to end distance

_R2(N)

> and the radius of

_gyration

_{R 2 (N)}

> . With rcm

being

the center of mass of a

_polymer

and ri the

position

vector of the i-th monomer

_they

are

_{given by}

Table 1

_gives

the results for the various densities and chain

_lengths

_considered,

while

_figure

4 shows the data for _p = 0.40 and

p = 0.80 for several chain

_lengths.

For ideal chains _one

_expects

làble I. -

values for

RG

> and

R2

>

_for

the various cases considered. The error bars

_{given for}

R2

> are estimated

_from

the

_fluctuation

between the 100

_{statistically independent}

_systems,

which

920

Fig.

4. -

Log-plot

of

R2

> and

_RG

> vs. N for p = 0.80

_(a)

and _p = 0.40

_(b).

The indicated

_slopes

of 2v =1 show that the

_expected

behavior is reached

_{quite clearly.}

R2

>

_/

4

> = 6 and

R2

_>

o:N211, V

=

1/2

[2, 20].

As can be seen from the

data,

for N = 100

_compared

to the

_expected

ratio

R 2>

is about 10% too small.

Sjrnilar

but

_stronger

effects were also found in

_[10].

This

_certainly

is a

_consequence

of the fact that the chains rather

_homogenously

fill a little

_fluctuating

disk,

while the random walk is a fractal

object displaying

self

_similarity

_[2].

The

_expected

_power

law with v =

1/2,

_however,

is fulfilled

very

well. From

scaling

for d = 2 _one would

_expect

R2(N)

>1/2=

Nvf

(N/p-2)

_(here

v =

3/4

the

_single

isolated chain value

_[2]

has to be

_{used). làldng}

the data from table 1 we find that the data for

_{f (x)}

_reasonably

well

_collapse

on a

_single

curve

_(besides

N =

20, 25),

however,

the

_slope

is not the

_expected

one. Ideal

_{scaling gives}

a

_slope

_{of -1/2}

for

_{f (x)}

in a

_{log log}

_plot,

while here the

_slope

is too

_large.

This can be

_explained

_by

the

_following:

_Scaling

is

_only

valid for the limits

x =

N / p-2 ’"

_const but N --+ _{oo, p -} 0. Hère we

_certainly

are out of this

_regime.

Similar effects

were also seen for d = 3

[21].

There is another

_interesting

_aspect

about how dense the

_systems

really

are. For

_regions

where the chains

_already

interact it makes sense to write

where all the excluded volume effects are taken into the variable

_{density dependent}

_persistence

length

_{p2 p (p)}

> 1/2 . Using (2)

we

_{get land}

_Ip

as function of

_density

as

_given

in table II. This decrease of

_lp

and

also £,

as also shown

_by

_figure

5,

_certainly

is an effect of

_{high density. Up}

to

7àble II. - Persistence

length

and mean bond

_{Iength for}

various

_{densities following}

the

_{definition of}

equation

(2).

For _p = 0

equation

(2)

requires

_Ip

= 00 since there the

_exponent

is v

_{= 3 /4}

instead

_of

1/2,

as is described

_by

the above-mentioned

_{scaling function.}

"

Fig.

5. -

(a)

Plot of the

_{persistence length}

_lp

vs.

_density

_p due to

_{equation (2). (b)}

Plot of the mean bond

length Î

vs.

_density

v.

This leads us back to the above discussion of the

_scaling

of

_{R2 ( N )}

> and

_{lîâ (N)}

> .

_Scaling

assumes that the internal structure of the chains do not

_change.

From the

data, however,

we see

that not

_only

the

_{persistence length}

varies with

_density,

but also the

_average

bond

_length

itself. The

_compression

of the chains is so

_strong,

that there is no

_space

for a

_selfavoiding

walk like

_blob,

which is

_required

for the

_validity

of

_{density scaling.}

Indeed,

if we normalize

R2

> not

_only

_by

N2"

but also

_by

£2

> at least for N = 100 between

p =

40%

and

p =

80%

the

expected slope

922

chains and the Rouse modes.

First,

we turn to the static

_scattering

function of the individual chain

where the

_{index 1 k}

_denotes

the

_spherical

_average over the orientation of k.

_Following

the standard

scaling

theories

_S(k)

should,

for d = 2 and v

= 4

for the isolated chain

[2], display

the fractal

scattering

law

here e

is the

_{typical screening length}

on which the chain still is

_expanded.

From

_{scaling e}

-vl(l-vd) _ -3/2 ,

however as one could see from the data of

R2

>,

R 2

> this

_screening

length

does not follow that

_power

law for the

_high

densities

_investigated

hère. Thus we

_hardly

can

_expect

the occurrence of a

k -4/3

_regime

for the

_highest

densities considered.

_Figure

6a shows

S(k)

for N = 100 and _p = 0.20

leading

to 03BE ~

30.

Going

back to

_figure

5 this is a reasonable

number,

giving

n =

(l212p/E2) ~- 110

monomers

_per

_{screening·distance.}

It means

_{for p}

= 0.20

even the

_largest

chains considered

_just

interact with each other. This also

_explains,

_why

there is no distinct

k-2

_regime

for _p = 0.20 rather than the

typical overshooting

effect of the

single

chain. The

_{overshooting, coming}

from the

_fluctuating

ends was discussed

_earlier[15]

and can also be found for d = 3. For d =

3,

since the chain can

_penetrate

_through

itself this effect is much weaker. With

_{increasing density}

this effect is more and more reduced.

_{Simultaneously,}

even the onset of the

k-4/3

_region

_disappears,

_indicating

that e

is

_{significantly}

reduced to about one or two

bond

_lengths

for _p = 0.80. The

sequence

of

figures

6a-c

nicely displays

this. For p = 0.80 we

_give

a

_scaling

_plot

of

_S(k)

vs. k for various N with the normalization that

_S(k

=

0, N

=

100)

= 1.

The data

_{reasonably collapse}

onto a

_single

curve.

_However,

_following

the

_snapshot

_pictures

of the introduction this is somewhat

_surprising.

There the cliains seem to

_{homogeneously}

fill

disk-shaped regions.

Such

disks, however,

would

_display

the

_(d

=

2)

Porod

_scattering

of

S(k) -

k-3.

Obviously

the

_shape

fluctuations are still

_strong

_enough

to

_prevent

such a

_scattering

behavior. In order to see this we have to consider the overall structure of the

_system.

With R2

>~

R 2

>N N the

_average

_{density throughout}

the chains

_approaches

a constant

for distances

Ar »

_e.

Thus for d = 2 a

_given

chain

_only

_directly

can interact with a constant number of

_neighbors.

For a dense solution of disks

_typically

one

_would

_expect

6

_{neighbors giving}

a

triangular

lattice like short

_range

order. If we assume the chains to form

_{s herical objects}

of radius r with

homogeneous

density they

would cover a disk of radius r =

2 12 RG.

Using

this,

we can

estimate the

_space

which would be

_occupied

_by

such chains.

_’Paking

the data of table 1 the

_chains

typically

would need about 220% for

_(N

_{= 100,}

_p =

0.80)

_to 150% for

(N

=

20,

_p =

0.20)

of the

space

available.

However,

this is

simply

impossible.

Therefore,

there must be

_strong

fluctuations of the chain

_shape.

_typically

we find the chains to have between 5 and 7

_{neighbors averaging}

to

6. This leads to a characteristic

_packing

_pictured

in

_figure

_7,

as

_expected

_by

_figure

2.

_Although

their overall

_density

is

_relatively

constant there must be

_strong

_shape

fluctuations. For ideal

ran-dom walks one has

_(d

=

2)

For the radius of

_gyration

one

_typically

_gets

a fluctuation reduced

_by

a factor of two

_[22].

Here we

Fig.

6. -

(a,b,c) Scattering

function

_S(k)

of the individual chains for various densities and N = 100 _as

indicated in the

_figures.

The data are

_averaged

over

_randomly

taken orientations of the

_scattering

vector k.

(d)’Scaling

plot

S(k,

N)/

_(1002/N2)

vs. k for N =

_{25, 50,}

100 _and

924

Fig.

7. -

Illustration of the _average local

_packing

structure.

which is close to the ideal random walk value.

_Although

the chains are

_{densely packed,}

this

_suggest

that these fluctuations allow for a

_{relatively high mobility}

of the chains. As we shall

_explain

in the

next

_chapter

this is the case. These fluctuations which indicate that the chains almost look like ideal

_chains,

also

_propose

that it

_might

be useful to

_analyse

the Rouse normal modes of a random walk. For a discrete

_system

_they

are

_{given by}

_[23]

’Iÿpically,

this one dimensional Fourier transform

_along

the

_sequence

of the chain is done with

periodic

boundary

conditions,

_{corresponding}

to a

_cyclic

chain.

_{By subtracting}

the second term we

cut the

_ring.

Note that we

_only

can

_expect

the Rouse modes to be

_eigenmodes

for ideal chains. In

principle

this would mean that the monomers

_arbitrarily

can

_pass

_trhough

each other.

_However,

we find the Rouse modes are

_eigehmodes

of the chains. Within the error bars the cross correla-tions

_Xp(t)Xq(t)

_>p#q

are zero. Even the

_amplitudes

of the modes follow the

_prediction

of the Rouse model

As

_figure

8 shows

_they

follow a common curve for _p = 80 and

p = 0.40 for the various chain

lengths.

The

_{expected slope}

is

_reproduced

_very

well and

_only

for small

_n/p2

_{values and p} = 0.40

significant

deviations can be found which is consistent with the results of

_S(k).

_{Forp large}

the

length

scale is reduced to the

_region

where the

_{self-avoiding}

walk behavior starts to dominate. There deviations are

_expected

and were also

seen

for d = 3

[24].

This also shows that for

p = 0.80

already

N = 25 reaches the

_asymptotic

regime.

In addition the ratio of the

_amplitudes

for the

two densities

_pictured

is the same as for the

_squared

radii of

_gyration.

Similar to the fluctuation of the radius of

_gyration

we can calculate

6z: =

_z

_X4

> -

X2

> 2)1/2

_/

X2

> . Since

_Xp

p p p p

as

_large

as the one for

_RG.

This is

_essentially

similar to the end to end distance

R2

>. For all densities and chain

_lengths

and

_p-modes

which follow the

_N/p2

_power

of the

_amplitudes

we find

0.8 b.,

1.0 in _very

_good

_agreement

to the above line of

_arguments,

_{demonstrating}

the overall

consistency

of the

_picture.

Fig.

8. -

Scaling

of the mode

amplitudes

Xp (N,

p)

> vs.

N/p2

for _p =0.80

_{and p}

=0.40. The ratio of

the two sets of data _agree with the ratios of

R2

> and

R2

> as shown in table 1 for

N/p2 >

5. The

date

give

the average over the Cartesian coordinates. The absolute value is somewhat smaller than

_expected

from ideal chains.

4.

_Dynamics.

After

_finding

that the static

_properties

of dense

_polymers

_very well

_compare

to ideal

_chains,

it is

_{especially interesting}

to check the chain

_dynamics.

Since the chain cannot

_entangle,

the

rep-tation model

_certainly

is not an

_{appropriate description.}

On the other hand in order to move

around the chains as well as the monomers have to

_get

_along

each other. This

_puts

_strong

con-straints onto the motions of the individual monomers.

_However,

how does this affect the overall

dynamical

behavior?

To

_investigate

these

_questions,

we first

_analyse

the mean

_square

_displacement

of the individual

monomers and the diffusion constants of the individual chains.

Figure

9 shows the data of the diffusion constant D for _p = 0.40 and

p =

_0.80,

as

_given

in

ta-ble III.

_Following

the Rouse model we

_expect

D =

KBTIN(,

where

kBT

is the

_{temperature and (}

the bead friction. Since there 1 - o

_temperature

in our

_system

but

_only

the

_acceptance

rate, we can

926

Fig.

9. - Diffusion constant 4D = lim

g3(t)

vs. number of monomers N. t->oo

this later in the context of the modes. What is

_surprising

is that the diffusion constant seems to

display

a weaker

_N-dependence

than the Rouse model

_suggests.

_Any

crossover to

_{reptation-like}

behavior would cause the inverse

effect,

namely

an

_{increasingN-dependence. Certainly}

we would

need a more extensive

_analysis

of a

_larger

_system.

As is showm

later,

such an acceleration does

not show

_up

in the

_meansquare

_{displacements}

of the monomers. Thus since for this estimate of D

_large

time intervals were

_used,

_figure

9

_probably

shows a residue of the

_{equilibration.}

Thus

Table III. -

Diffusion

constants 4D

_for

various N and

_{density p}

= 0.4 and

we think that the data are consistent with D -

N-1.

_Figure

10

_gives

_examples

for the motion of individual monomers N = 50 and _p =

0.40,

0.80.For N = 100 the data show the same behavior.

Fig.

10. - Mean

square

displacement

gl(t)

and

_g2(t)

and the diffusion of the

center

of mass of the chain

g3(t)

for p

= 0.80

_(a)

and _p = 0.40

_(b)

and N = 50. The two sets of data for

_{Yl/ Y2}

_{give the}

results for middle

(lower data)

and end monomers

_(upper

_data).

The difference in the motion between inner and the outer

monomers decreases with

_increasing

_density.

What is

_especially

_striking

is that the difference between the mean

_square

_{displacements}

_{gl , 92} of

the total

_chain,

_{given by}

and

_{corresponding}

data of gl, 92 for individual inner monomers is much smaller than for d = 3. For d = 3 this effect is

quite

dramatic since this shifts this

visibility

of

reptation

in these

quatities

far

_beyond

the

_{entanglement length}

_[23-25].

_However,

similar to the chain confined to a

_straight

tube,

g2-for-inner

monomers saturates much earlier than the mean

_square

_displacement

_gl. Such

an effect was first observed for the chain in the

_straight

tube

_[26]

and seems to be

_typical

for such constrained

_geometries.

This

_certainly

must be an effect of the

_segregation

of the chains and the

confinement due to the

_{surrounding compared}

to the 3-d case. For both cases we find a nice

t112

behavior for _{gl, 92 up} to the

_longest

relaxation time of the chain. After that time the diffusion of the whole chain takes over. Table IV

_gives

a collection of relaxation times. So far all data seem to

reasonably

agree to the Rouse model. The most crucial test for the

_{dynamic properties}

is the

928

Thble IV. -

Relaxation times Tp

for the first

three

_{modes p for}

chains at densities

_of

_p = 0.4 and 0.8.

The error bar

of

the data is around 15% .

rates of the Rouse modes

_Xp

as well as the autocorrelation function of the radius of

_gyration

squared. Figure

11 shows the relaxation function

for N =

50,

_p = 0.40 as an

_example

for the first three modes.

_{They display}

a

_good

_single

exponen-tial behavior for

_longer

times,

which enables us to calculate the relaxation times which enables us

to calculate the relaxation times _{Tp .} For the Rouse model one

_expects

Figure

12

_gives

a

_{scaling plot}

of the relaxation times _7p from table IV.

_They

all follow the

_single

curves and

_roughly

follow a

_slope

of

_(N/p)2.

From these data as well as from the diffusion

con-stants D we

_directly

estimate the normalized bead friction

_{(IKBT. Using}

the data of table II for l and

_lp

and table III for D the two friction constants do not _{agree very} well. We find that the relaxation of the individual modes is

_considerably

slower than one would

_expect

from the diffu-sion. For d = 3 melts it was found

_[25]

that

_they

_agree

_very well.

_Taking

_e.g.

N =

_50,

p = 0.8

(here

the statistic is better than for N =

100)

the modes

_give

(IKBT

6.4 _x

102

while D

_gives

1.5 x

102,

_giving

a ratio of ~ 4.3. For _p

= 40%

this ratio is reduced to a value of less than 3. Thus with

_{increasing density}

this deviation increases as well.

However,

this reflects that the monomers

cannot

_{interpenetrate}

_freely,

as

_required

_by

the Rouse model.

_Thus,

_although

the Rouse modes

are

_eigenmodes

and ₉₁ does not show

_significant

deviations from

_Rouse,

the diffusion seems to be

governed by

shape

fluctuations,

which are faster

_by

a constant factor than the internal

reorgani-zation which would allow the Rouse modes to relax. Similar distinctions in time scale have been

proposed

for star

_polymers

_[29].

Finally,

let us check the overall relaxation of the radius of

_gyration.

The

autocorrélation

function shows a

_single

_exponential

_decay

similar to the modes. A more detailed

_inspection

shows that

rouphlv

In order to obtain the

_precise

N and p

_dependence

much more estensive simulations are needed.

The relaxation times are

_given

in table V. Since the fluctuation of

_RG

are the

_shape

fluctuations

Fig.

11. -

Examples

of a

_typical

mode relaxation

_plot

for N =

_{50, p}

= 0.4. The time is

_given

_{in Monte}

Carlo

_steps

_(mcs).

Fig.

12. - Relaxation

times of the first three Rouse modes vs.

N/p

for _p = 0.40 and _p = 0.80 and different

930

Thble V. - Longest

relaxation times

_of

the auto-correlalion

_function

R$(t)Rl(0)-(flÕ(0»)2

»

_/(

RG

>2 -

(Rô)2

_>)

_for

various

chain

_lengths

and densifies _p = 0.4 and

0.8. The

N2

_dependence

is

_{rougihlyfulfiued.}

The error bar

_of

the data is

_expected

to be between 10% and 15%.

Fig.

13. -

Acceptance

rate

_A(S)

of

_attempted

moves vs.

_density

_p. The insert

_gives

the normalized devia-tion of the

_acceptance

rate

_A(S)

with

_respect

to the

_acceptance

rate  =

A( p

=

0)

of an inner monomer of

a

_single

isolated chain.

mode.

_Using

_again

_{equation (11)}

with p = 1 shows that we arrive at a bead friction which is in

very

good

agreement

to the diffusion

data,

while the modes

_give

a different result. This rather

regular

behavior indicates that there are no

_precursors

of the

_glass

transition

_yet

in

_agreement

to the mode

_analysis.

The

_density

_dependence

is

_directly

seen also from the

_acceptance

rate of the moves. For lattice

_gases

it is known that due to

_back/hopping

corrélations the

_acceptance

rate

_decays

slower than the increase of relaxation times

_suggest

_[27].

_Figure

13 shows the relative

power

law with

density

as for

_THe;

_indicating

that back

_hopping

_only

enters in the

_prefactor

and is dominated

_by

the intrachain correlation of monomers

_{nearby along}

the chemical

_sequence

of the

polymers.

5. Conclusion.

We

_presented

a detailed

_study

of the

_properties

of a melt of two-dimensional

_polymer

chains.

By

the use of the bond fluctuation method we were able to

_study

both static and

_{dynamic properties}

of the chains.

For statics we found that the

_chains,

_{although they}

cannot cross each

_other,

_display

_typical

random walk behavior for

_{many uantities}

like

R2

>,

RG

>,

_{S( k)}

and the Rouse modes. As function of

_density

the

_decay

of

_{R ,}

_Rb

shows that we are

_explonng

the hmit of

_high

_density

melts. The chains

_segregate

_comptetety

and the ends seem

_only

to ftuctuate on small scales. Nevertheless the overall

_shape

fluctuations turn out to be sufficient to result in a

_scattering

law

_{S(k) -V k2}

instead of the Porod

_scattering.

The

_amplitude

of these

_shape

fluctuations is of the same order as for

ordinary

random walks. For the lowest

_density

considered we

_nicely

find the inner

k-4/3

_regime

for short internai distances. This is

_important

with

_respect

to recent ideas of the

_e-collapse

transition of

_{polymers [28].}

_Considering

the

_present

result for these

_concepts

the disorder of the defects in the

lattice,

althougt they

are

_annealed,

is crucial. Otherwise the

_scattering

function of

our

_systems

would show a

_very

different behavior.

_Although

the overall

_properties

of the chains

are the same as for ideal Gaussian chains there are some distinct differences with

_respect

to the inner structure, such as the uniform

_density

in the chains or

_probability

of monomers

_approaching

each other. The latter

_quantities

have been calculated

_[1]

and a future

_{investigation}

should be able

to check this.

The most

_unexpected

results were found for the

_dynamical

_properties.

The Rouse model does not consider

_any

other interaction betweeen the monomers than the chain

_{connectivity.}

Consid-ering

that this

_requires

monomers

_crossing

each other

_freely,

we found it rather

_surprising

that the

2-melt almost

_perfectly

_reproduces

the Rouse model. There is no

_sign

of decrease of

_mobility

due

to

_topological

constraints. Ibe

_only,,

but

_significant

deviation is

_given

in the time scales.

_Although

the mode relaxation as well as the diffusion

_give

the same N

_dependence

of the relaxation

time,

there is a difference in the

_prefactor

of about 4

_{for p}

= 0.8. It shows that diffusion via

_shape

fluc-tuations is faster than internal structure relaxation. This

_suggests

that the

_similarity

to the Rouse model is rather accidental since the 2-d

_exponents

for the dense

_system

and the related

_quantities

exhibit the same

_power

law as the

_ordinary

random walks. We therefore think that it is

_especially

interesting

to

_investigate

the chain

_dynamics

of a

_collapsed

3-d chain. _{There many}

_interesting

de-viations

_might

_occur, while for the

_single

_collapsed

2-d chain we do not

_expect

different behavior

compared

to our

_present

_findings.

Acknowledgements.

932

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