Enrichment of the chain ends in polymer melts at interfaces

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Enrichment of the chain ends in polymer melts at interfaces

Jiang-Sheng Wang, Kurt Binder

To cite this version:

Jiang-Sheng Wang, Kurt Binder. Enrichment of the chain ends in polymer melts at interfaces. Journal

de Physique I, EDP Sciences, 1991, 1 (11), pp.1583-1590. �10.1051/jp1:1991226�. �jpa-00246437�

Classification

Physics

Absiracts

61.40k 68.10

Enrichment of the chain ends in polymer melts at interfaces

Jian-Sheng Wang

and Kurt Binder

Institut fur

Physik, Johannes-Gutenberg-Universitfit, Staudinger Weg

7, D-6500 Mainz,

Germany

and Max-Planck-Institut fur

Polyrnerforschung,

D-6500 Mainz,

Germany

(Received 21 May 199J, accepted in

final form

8

July1991)

Abstract. By Monte Carlo simulation of _two lattice models of

polymer

melts, the enrichment of the chain ends in the

vicinity

^of a wall is observed. For the bond fluctuation model 4ith attractive

interactions between the ends and wall,

s/ka

T

=

I and 0, the enhancement of the

density

of ends increases with chain

lengths,

while when the interaction is

repulsive, s/kB

^T ₌ + I, the ends get

depleted.

The

self-avoiding

walk model was simulated with the

slithering

snake

algorithm

and behaves

similarly.

The

polymer

center of _mass

density profile,

radius of

gyration

and end-to-end

distance components

parallel

and

perpendicular

to the wall _as functions of center of _mass distance

z scale with

R~

where R~ ^{is radius of}

gyration

of the

polymer

chains in the bulk.

1. Introduction.

The strtJcture of _a

polymer

melt in the presence of _a wall has

important

consequences for many interfacial

properties [1, 2].

^{This includes surface tension and}

dynamical

_processes

which _occur at interfaces. Recent

results,

not

explained by theory,

of

wetting

transitions in

binary polymer

mixtures

[3]

_may be related to the structure of

polymers

close to walls. There have been theoretical studies

[4, 5]

and many

computer

simulations

[6-14]

_on the strtJcture of

polymers

at interfaces. These studies reveal that the

polymers

_{near a} wall _no

longer obey

Gaussian statistics

[9, 13]

even if

they

do in the bulk. The

polymer

chains _are flattened _out like _a

pancake

and

almoit

two-dimensional.

The other

interesting

effect due to the wall is that the ends tend to

stay immediately

close to the wall. The

density

of the ends would be 2

~P/N

if there is _no distinction between end

monomers and _monomers in the middle of _a

chain,

where ~P is the _monomer

density

and

N is the chain

length.

An enhancement of the

density

of ends is observed in both lattice model

[8]

and continuum model

[9-1ii

simulations. This effect is

purely entropical

since there is _no additional interaction between wall and _monomers besides exclusions for the models studied.

While ten Brinke et al.

[8]

found that the enhancement is

independent

of the chain

length

for _a cubic lattice

model,

Kumar _et at.

[10]

^{did find} _an ^increase of the enhancement with

N,

for N

=

50,

100 and

200,

in _a continuum

polymer

model. Kumar et al.

suggested

exponential

saturation to a finite value for

large N,

p~mc+aexp(-b/N), (1)

1584 JOURNAL DE PHYSIQUE I M II

where p~ is

N/2

times the fraction of ends to total _monomers at the immediate

vicinity

of the wall

(see

definition

Eq. (3)).

Recently,

de Gennes

[2]

considered three

possible

situations of the end

effects, depending

on the interaction

strength

between the end _group and the surface

(wall).

Let _u

= ha ~/k~

T,

where A is _excess surface free _energy, if _an end is at the

surface,

_a is the lattice

spacing, k~

is the Boltzmann constant and T the

temperature.

(a)

For _u « I the chains _are ideal. Attraction between -the ends and the surface is weak.

There will be _no

enhancement,

_p~

m I _;

(b)

Intermediate attractive energy, um I. The ends which _were

originally lying

in _a thickness of radius of

gyration, _Ao

= N ~/~a, are

trapped

at the

surface,

_so that p~ m N ~/~;

(c) Strong attraction,

_{u »} I. The chain ends _are

pinned

at the surface and the

polymers

_are stretched. Even in this _case the enrichment is also _p~

m

N~'~

An

interpolation

between _case

(a)

and

(b)

is

proposed

_as

~~ l ₊

~~~/J§

^{' ~~~}

where _a and b _{are some} constants

independent

of N.

In this work _we take _a careful look at the N

dependences

of various chain

properties

_near

walls, using

_a bond fluctuation lattice model

[15].

This model combines the

advantages

of

both _more realistic off-lattice

polymer

models and

simple

lattice models with _a faster

updating speed;

for _more motivation and

details,

_see references

[3, 15-17].

One of the

objections against using

_a lattice model _was that the

oscillatory density profiles

_{near a} wall _cannot be observed

[9]. However,

the bond fluctuation model does have the

oscillatory profile

at

high

volume fraction similar to those observed in

continuous-space

simulations

[9-13].

The

enrichment of the ends in the bond fluctuation model _was found to increase with

N when there is _{even no} interaction between the wall and the

ends,

similar _to the continuous models. We believe that the bond fluctuation model

captures

many

important

features of the

continuous-space

models.

The bond fluctuation model has _a slow

dynamics comparing

to the

slithering

snake

(reptation) algorithm [18,19],

which limits _us from

simulating

_very

long

chains. The relaxation time of the bond fluctuation model in _a dense melt is

roughly

_r

^N~.

This

gives dynamical

behavior consistent with real

polymer systems. However,

it makes

equilibration

difficult for

long

chains. We thus also studied the

slithering-snake

cubic lattice model and

compared

the results.

Following

reference

[8],

_we make

slithering~snake

_{moves as} well _as

kink-jump

and crankshaft _moves. For the

slithering

snake

algorithm

the relaxation _{goes as}

r ~

N. One Monte Carlo

step

here is defined _as the number of _moves

equal

to the total number of _monomers in the system, so that the

computer

time per step is

independent

of the chain

length

for _a

given

^volume fraction. In this model the number of ends _are 8 times _more

comparing

to the bond fluctuation model for the _same system

size,

thus _we have _more statistics.

Indeed,

the results _{are more} accurate.

2. Simulation method.

We outline

briefly

the bond fluctuation

model,

for _more detail _see references

[3, 15~1?].

The

polymers

_are

represented by

chains of

length

N _{on a} lattice. The

lengths

between consecutive

monomers are allowed to take _a set of values between 2 and

fi.

_Each

monomer

occupies

8

sites of _a cube of dimension 2 _x 2 _x 2. We consider _an athernlal

system

: there is _no other interaction

except

mutual exclusion of the _monomers. Two walls _are located _{at z}

=

0 arid

H+ I in _a system ^{of dimension} Lx Lx H. We used

mostly

system ^{of size H} ₌

40,

L

=

40,

with half of the sites

occupied by

monomers

(~P

=

1/2).

Previous

study

^has shown that this

density corresponds

to a dense melt

[17].

The walls _are

impernleable.

An interaction

energy, s, is introduced if the ends _are

immediately adjacent

to the walls. This parameter is

quite

natural to consider since the

polymer

end groups can differ from the inner groups. Three

cases were

considered, s/k~

T

=

0,

±1. In the _x and y directions _{we use} helical

boundary

conditions _or

periodic boundary

conditions. In the _case of helical

boundary conditions,

the

sites _are numbered from I to Lx Lx H. The

neighbors

of site I _are I-

I,

I+

I,

I-L, I+L, I-L( I+L~. _However,

_{if these numbers}

are outside the _range I to

L~x _H, they

_are not allowed to be

occupied.

The helical

boundary

condition

simplifies programming considerably.

But it does introduce unfavorable finite-size effect of order I

IL

due to

anisotropy.

With the helical

boundary condition,

_we have achieved _a

speed

of

5 _x 10~ _{monomer moves per} second _{on a} FUJITSU VP100 vector computer,

by simulating

simultaneously

50

independent systems.

^The

length

of _a Monte Carlo _rtJn is

typically ¹⁰⁶

steps

per monomer. The results _are

averaged

_over 4 to 8

independent

_runs. The

length

of the chains

N is limited

by

the relaxation time. This limit _{us to}

investigate

_systems with N

~ 80 in order to achieve

equilibration.

3. Enricbment of tbe ends _near tbe wall.

First _we

present

^the monomer

density,

~P

(z),

_{as a} function of distance _z from the wall. The

density

has _an

oscillatory profile,

_see

figure

I. This oscillation has been observed in the studies of references

[3, 20].

It _occurs at

relatively high

volume fraction of the _monomers. For low volume fraction _or semi~dilute solutions the

density

_near the wall is lower than that of the bulk

[21].

Such oscillations _are absent in the fixed bond

length (one

lattice

spacing)

model.

They

in fact resemble the oscillation in off4attice models

[9-13]

and _are similar to

simple

fluids

near a hard

wall,

due to

packing

of the

particles against

the wall. The

amplitude

of oscillation

decays

and the bulk uniforn1value is reached after 8 lattice

spacings.

The

profiles

near wall

depend only weakly

_on chain

length

N.

o.7 1.4

e

' 1.3

0.6

,

_

1.2

o

h4 .

~

j~

*

.

° & l-I

e '

~'

. , i ; ^{' '}

. ,

z

z

Fig.

I.

Fig.

2.

Fig.

I. Monomer

density 4~(z)

_as a function of distance _z away from the wall, for chain

length

N =10 (o), 20 (li), and 40

(+).

There is _no interaction between

polymer

ends and the wall (s

= 0).

Fig.

2. Normalized

density

of chain ends,

p~(z),

as a function of distance _{z away} from the wall, for chain

length

N

= 10

(o),

20

(li),

40

(+),

and 80

(D).

1586 JOURNAL DE

PHYSIQUE

I M II

We consider the normalized

density

of chain ends

Pe(Z)

"

( $

,

(3)

where

~P~(z)

is the average number of

polymer

ends at

layer

z and

~P(z)

is the number of

monomers at

layer

_z, all normalized

by

the surface _area

L~.

The

quantity

should be I if _no

enrichment _or

depletion

_occurs. It tums out that ~P

(z)

also

weakly depends

on N.

However,

whether _we consider the absolute

density

of ends

~P~(z)

_or consider the ratio

p~(z),

_our

conclusion

conceming

the

large

N behavior is the _same. In

figure 2, p~(z)

is

plotted

for

polymer lengths

N

=

10, 20,

40 and

80,

as a function of _z, for the _{case s}

=

0. It is

quite interesting that,

_even for _s

=

0,

the ends _are enriched at _z

= I and the

density

increases with N. Note that since the number of total ends is

conserved,

_an enrichment _near the wall leads _to

depletion

_away from the immediate

vicinity

of the wall. The

region

of

depletion

becomes

larger

_as N becomes

larger,

consistent with the notion that the affected

region

should be

proportional

to the radius of

gyration

in the bulk. Our

results, presented

in

figure 2,

_are in

qualitative

_agreement with that of reference

[10]

for _a continuum model.

The

largest

effect _occurs at _z

=

I,

monomers in contact with the wall. The normalized

density p~(I)

is

plotted

_vs.

^N'/~

for

s/k~

^T ₌

0,

± in

figure

3. In the _case

s =

0,

_{we see a}

good

linear

dependence

for N ~10.

However,

the

slope

is very small

(0.01).

It is not clear that this is the trtJe

asymptotic

behavior. lvhen the interactions between wall and ends _are

repulsive,

the ends get

depleted.

The

depletion

should become

independent

of the chain

length

^for

long

chains. For

s/k~

^T

=

I,

where the wall is

attractive,

the enrichment is much

higher.

It increases but slower than

N~/~.

The data _appear consistent with _a saturation to _a finite value for

large

N

according

to

equation (2).

This is shown in

figure

4

by plotting

I

/p~(I )

_vs. N ~/~.

Similar results for the

slithering

snake model at bulk volume fraction ~P

=

1/2

_are obtained for

s/k~

T=

0, 0.5,

1, and 2

(see Fig. 5).

The volume fraction ~P

=1/2

is used for

3 4 5 6 7 8 9 0.1 0.2 0.3 0A 0.5

N~'~ N~l'2

Fig. 3.

Fig.

4.

Fig. 3. Normalized density of the ends at _z

= I, _p~(I), as a function ofN~'~ _{for the ends} interaction with the wall

s/ka

T ₌ I

(li),

0

(+),

and I

(o).

Fig.

4. Normalized

density

of the ends at z = I for

s/k~

T

= I,

plotted

I

/p~

vs. N ^~'~as

suggested by

equation (2).

0.8

W

' Ck

N4

Fig.

Fig.

op to s/k~ T ⁼ - 2, - 1, -

1588 JOURNAL DE

PHYSIQUE

I M 11

u with

s/k~

T. The data fit _a

straight

line

well,

in accordance with

equation (2)

for the limit N - cc. This result

suggests

^{that the ends in the}

long

chain limit behave

approximately

like ideal _gas. The enhancement of the ends is

simply proportional

to the Boltzmann factor

exp(- s/k~ T).

We also _test whether the N and _s

dependence

of the enhancement _can be casted in _{a more}

general scaling

form with the

scaling

variable

N~/~e~~ (cl Eq.(2)).

A

scaling plot N~/~/p~

vs. N ~/~e~~ is

presented

in

figure 8, identifying

_u with

s/k~

^{T. It} appears that such _a

scaling

form holds for

large is _(. Systematic

deviations for small _s may be due to

equating

u with

s/k~

^T

directly,

since

strictly speaking

u is _a free energy also

containing entropic

contribution. Note that the curvature for small

N~'~e~~

_indicates

a deviation from

equation (2).

Assuming

that

p~(I)

saturates to _a finite value for

large

N but in _an

exponential

fornl p

~

l

i

= c + a exp

(- NiNoi

,

(4)

we can also fit the results well. The data _appears to fit less well with

equation (1).

4.

Scaling

behavior of the

prorJles.

The

following

set of

quantities

_are calculated _{as a} function of the z-coordinate

(truncated

to _an

integer)

of the center of _mass of _a

polymer:

the distribution of the center of _mass,

W~(z)

I-th

component (I

= x, y,

z)

of the mean-square radius of

gyration, R(,

,

(z)

=

jj ( _q,

, r~,

,

~)

;

(5)

J i

and the mean-square end-to-end

distance,

~/(Z)

"

(~N,I ~l,I)~)

,

(~) respectively.

In the

above,

_q,, is the I-th component ^{of the}

position

of the

j-th

_monomer on

the chain and r~~,, ^{is the I-th} component ^{of the}

position

of the center of _mass. Because of the symmetry ^between x and _y

direction,

_we take the _average of the _x and _y

components.

^We are

particularly

interested in their

scaling

behavior for

large

N. Since there is _no other

length

scale

except

^{the radius of}

gyration

in the

bulk, R~

=

R(

~ ⁼

R(,

=

R(,

we

expect

^{that when}

'- 3

z is scaled

by R,

and the

quantities properly

nornlalized

by

R to certain power, we could obtain _a universal

scaling

curve, as has been

suggested

in references

[6, 10].

Thus _we consider the

following scaling

relations _:

Nw~(z)

₌

f~(z/R) (7)

JG[;

(z)/R~

=

f(z/R)

,

(8)

where

f

is _one of the four

quantities

mentioned above.

The

scaling plots, figures 9,

10 and

it,

confirm the

expectation.

The end-to~end distances

(not shown)

behave

similarly

_as the radius of

gyration.

The _x component ^{increases when the}

polymers approach

the

wall,

while the _z component ^{decreases to} zero. That

is,

the chains

become flat and

pancake

^like in the immediate

vicinity

of the

wall,

as has been observed in other models

[6-13].

There is _a very weak

peak

for the _z component before

finally decaying

to

the bulk value. The

peak

is _more

pronounced

for _s ~0

(see Fig,12).

This _means that

polymers

_some distances away are stretched in the normal direction of the wall. This effect has not been noticed before.

~ "O

O

"

OO h

O O

CQ O,°

~ "

O O

m

lE

°O

O °~ 'O~i

'04 _~

" '~ '

i

fQ Cl O

p

~

8

~

~ (

°

" °"° '

~

0

zlR

Fig.

9.

Fig.

10.

Fig. 9.

Scaling

of the center of mass distribution, with chain

length

N

= 10

(o),

20

(li),

40

(+),

and 80

(D).

Fig.

10.

Scaling

of the x-component of mean-square radius of

gyration,

with chain

length

N

= 10

(o),

20

(li),

40

(+),

and 80

(D).

o~

~~»~," 'o(

,

9 ,,o o

"" ' "

"»

m 0.8 ^"~

Ct _,

'

~ ^~6 o~

r~

'~~ ~

,

Ct

O~

~

0 2 3 4 5 5 lo 15 20

zlR

_z

Fig.

il.

Fig.

12.

Fig.

ll.

Scaling

of the z-component ^of mean~square radius of

gyration.

Same chain

lengths

as in

figure10.

Fig.

12. The _x

(o)

and _z

(+)

components ^of mean~square radius of

gyration

for N =16,

s/ka

T= -1.

5. Conclusion.

We have demonstrated

by

Monte Carlo simulation that the enhancement of the ends near a

wall does _occur. We introduced the interaction between ends and wall and studied its effect.

In both models the enhancement of the

density

of ends is consistent with saturation to _a finite

value for

large

N. An

N~/~ dependence

has not been

clearly

_seen for the models and

1590 JOURNAL DE

PHYSIQUE

I bt II

parameters

simulated,

_even for the _case

s/k~

T= -2. It could be that _our values of N _are still too small to be in the

asymptotic region

of the N ^~'~ behavior _or it is

only

obtained in the strong interaction limit

(s

- cc

).

The _range of disturbance due to the wall _on the center of _mass of the

polymer chains,

radius of

gyrations

and end-to-end distances extends to a distance of _{one to two} bulk radius of

gyration

of the

polymer

chains in the melts.

Acknowledgement.

J.~S. W. is

supported by

the Max~Planck~lnstitut fur

Polymerforschung,

and K.B.

by

Materialwissenschaftliches

Forschungszentrum

der Universitfit Mainz

(MWFZ~.

Part of the

computation

_was carried out at the Kaiserslautem

Regional Computer

Center

(RHRK).

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