Flexible polymers in a nematic medium: a Monte Carlo simulation

HAL Id: jpa-00247857

https://hal.archives-ouvertes.fr/jpa-00247857

Submitted on 1 Jan 1993

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Flexible polymers in a nematic medium: a Monte Carlo simulation

J.H. van Vliet, M. Luyten, G. ten Brinke

To cite this version:

J.H. van Vliet, M. Luyten, G. ten Brinke. Flexible polymers in a nematic medium: a Monte Carlo simulation. Journal de Physique II, EDP Sciences, 1993, 3 (5), pp.603-609. �10.1051/jp2:1993154�.

�jpa-00247857�

Classification Physics Abstracts

61.25H

Short Communication

Flexible polymers ⁱⁿ

_a

nematic medium:

_a

Monte Carlo simulation

J.H. _van

Vliet,

M-C-

Luyten

and G. ten Brinke

Department of Polymer Chemistry, University of Groningen, Nijenborgh 16, 9?4? AG Groningen, The Netherlands

(Recdved

22 June1992, reveised 9 March 1993, accepted 15 March

1993)

Abstract. Monte Carlo simulations of self-avoiding random walks surrounded by

aligned

rods _{on a square} lattice and

a simple cubic lattice _were performed to address the

topological

constraints involved for dilute solutions of flexible polymers in _a highly oriented nematic sol-

vent. The nematic constraint exerted by the solvent is modelled by

requiring

that the distance

between nearest

neighbour

chain segments ^{in the direction} of the director of the nematic solvent is such that _a discrete number of rods, representing the perfectly ordered nematic solvent, fits

exactly in between. Rod lengths of _{one or} two lattice spacings _are used. In this _case interesting scaling behaviour for the radius of gyration and the end-to-end distance is found. Perpendicular

to the director the coil behaves _{as a} two-

(or

_one- in the _case of _{a square}

lattice)

dimensional self-avoiding walk, whereas parallel to the director it behaves

as a random walk. The confor- mations have _a disk-like shape with the radius of gyration perpendicular to the director being considerably larger than the parallel component. These simulations address only _one aspect of flexible polymers in nematic solvents. In reality the _presence of the coil also reduces the entropy of the solvent which has exactly the opposite effect _on the shape of the polymer coil.

Introduction.

An

interesting

area of

polymer physics

_concerns ^the size and

shape

of

polymer

coils in _an

anisotropic

environment. A

simple example

is _a dilute

polymer

solution confined between _two

parallel plates.

In this _case the intrinsic

anisotropic

conformation of the coils interfers with the

spatial

constraint. When the distance between the

plates

is

reduced,

the coil first orientates with its two

largest

_axes

parallel

to the

plates.

The deformation of the

polymer

coil

only

starts when the distance is

comparable

to the _span

length

in the direction associated with the shortest axis

ill.

A nematic solvent constitutes another

example

of _an

anisotropic

environment for the dis- solved

polymer. Experimentally,

the conformations of flexible

polymer

coils in _a nematic solvent

were studied

by

NMR and

small-angle

neutron

scattering (SANS).

NMR

gives

information _on

604 JOURNAL DE PHYSIQUE II N°5

the local

order,

whereas the

global

orientation of the

coil,

described

by

the _square radii of

gyration parallel, S(,

^and

perpendicular, S[,

to the director of the nematic solution _can be found

by

SANS [2]. ^{Besides these}

microscopic techniques,

the

global

orientation of the coils has also been

investigated by viscosity

measurements,

interpreting

the results _on the basis of

a model

presented by

Brochard

[3-5].

The

interesting point

is that under certain conditions the

polymer

^coil _appears to be

elongated parallel

_to the director of the nematic solvent. At first

sight

this situation resembles that of _a

polymer

coil confined between _two

parallel plates, suggesting

that also here the effect is

merely

_an orientation of the

anisotropic shaped

coil [6].

This observation

prompted

us to

study

the

shape

^of a

polymer

coil in _a nematic

solvent, using

lattice simulations.

So far the

experimental

results for

(semi-)flexible

chains in _a nematic environment have been

interpreted using

theories in which _a mean-field

potential

is introduced. The

potential

models the

coupling

of the nematic field with the chain.

[7-14]. However,

the effect of the excluded volume has been

largely ignored.

The Monte Carlo simulations for lattice

chains, reported here,

are

explicitly

undertaken to address this aspect. In these simulations the nematic solvent molecules _are modelled _as rods

aligned perfectly parallel

to the director.

Therefore,

it is obvious that the relevance of _our simulation is limited to _a temperature

regime

far below the critical

temperature ^{for the nematic} to

isotropic phase

transition of the solution.

Simulation method.

Monte Carlo simulations of

self-avoiding

random walks

(SAW) representing

the flexible

poly-

mer, on a 100 _x 100 _x 100

simple

cubic _or 100 _x 100 _square lattice _were

performed.

The

lengths

N of the lattice chains

considered,

were 24r149 steps. ^A step is defined

as a lattice _vector

connecting

two lattice

points occupied by

connected chain segments; so a chain of N steps

comprises

N + I

segments.

^{One of the lattice} axes is chosen to be

parallel

to the director of the nematic rods. The number of chain conformations is restricted

by

_a nematic constraint.

The constraint is modelled

by requiring

that the distance between

nearest-neighbour

segments in the direction of the director

(here

the _z

axis)

is such that _a discrete number of rods fits

exactly

in between

(Fig. I).

It

resembles,

but differs in _an essential _way from the model of SAW's _{on a} lattice in _a field of fixed

asymmetric

obstacles studied

by Baumgirtner

et al. [15]

and Dimarzio

[16].

Conformational _averages of

quantities

A _are calculated

as

IA)

₌

f

A;

(1)

where the

angular

brackets indicate _a conformational _average, I is the conformational index and _n the total number of conformations.

The

reptation algorithm

is used to generate different SAW conformations. The first 400,000

attempted

_moves after

completion

of chain

growth

_were

ignored.

From the

subsequent

inter- val of

40,000-400,000 attempted

_moves ^the

quantities

of interest _were calculated each time _a

conformation

satisfying

^{the nematic constraint} was obtained. This

procedure

_was

repeated

for at least _seven

independent

_runs. After

averaging again,

the statistical _{errors were} calculated

as usual from the _{averages per run.} We _were able to generate ^reliable results for chain

lenghs

up to N

= 150.

During

the simulations the segments were at

regular

intervals rotated _{over an}

angle

of 90° around _one of the

randomly

chosen

orthogonal

lattice _axes with the

origin

of the coordinate system in the centre of the lattice. The centre of _mass of the SAW is maintained

near this

origin by

_a translation of the whole conformation when necessary. This rotation

i

Fig. 1. Snapshot of _a chain conformation _{on a square} lattice for N

= 50, after 440,000 attempted

moves:

(e-e)

chain segment;

(o-o)

rod; _n is the director of the nematic rods. Length of rods is _one

lattice spacing.

ensures a more efficient

sampling

of the relevant part ^{of the}

configuration

_space

(a

kind of

pivot algorithm) iii].

Results and discussion.

The averages for the radius of

gyration parallel, (Sj()

=

(S))

and

perpendicular, (S[)

=

((S])

₊

(Sj)) /2,

to the

director,

and the

corresponding

mean-square end-to-end

distances,

(R(), and, (R[),

_were calculated for the _square and

simple

cubic lattice. Two different

lengths

of the

rods,

_one and two lattice

spacings long,

_were chosen. The

scaling

exponents for the square radii of

gyration

and for the _square end-tc-end distances _were obtained

by

linear

regression according

to

(In (A;))

= In

(C;)

₊

2v;In(N) (2)

where A; is _now the

parallel

_or the

perpendicular

component ^of the _square radius of

gyration

or square end-tc-end distance. Since the scatter in the data used for linear

regression

is

comparable, only

_one characteristic

example

is shown in

figure

2.

A

typical

chain conformation is shown in

figure

I. Results _are

presented

in tables I and II.

For all chain

lengths

^{considered the}

perpendicular

component of the radius of

gyration,

^and the end-to-end distance is

larger

^{than the}

parallel

_one.

Moreover,

these results show that _a chain

subjected

to _a nematic constaint _{on a}

simple

cubic of square lattice behaves

as a SAW in the

plane,

_{resp. on} the

line, perpendicular

to the

director,

and _{as a} random walk in the

parallel

direction. The SAW

scaling

exponents are the

Flory

exponent, _vF, ^for a

spatial

dimension d'

= d

I,

d

being

the lattice dimension considered.

Hence,

lAi)

~

N~~~;

_1/F

= ~, ~ ⁼

)

13)

It is also clear that the values of the

scaling

exponents are within _error

independent

of the

lengths

of the rods. This result _can be

easily

understood _on the basis of _a characteristic conformation like the _one

presented

in

figure

^I. A disk-like conformation with its

largest

606 JOURNAL DE PHYSIQUE II N°5

<Lnls~,~l>

0

~ LnlNl

Fig. 2. Double

logarithmic

plot for determination of critical exponents, for _a SAW _{on a} simple cubic lattice:

(e) (In (S [) _{(o) (In (Sj()} Length

of rods is _one lattice spacing. Maximum relative _error is 0.04.

Table I.

Scaling

exponents.

s2 ~2

rod

lattice

length* 2vi 2vi

square I 1.94 + 0.02 1.09 + 0.03 + 1.00 + 0.04

cubic I 1.44 + 0.05 0.93 + 0.05 1.44 + 0.06 + 0.06

cubic 2 1.44 + 0.03 1.03 + 0.02 1.47 + 1.00 + 0.03

* unit of length is _one lattice spacing

Table II.

Average

size.

/~

~2 ~2

1 ((

~

25 3.5 2.5 0.03

50 8A 4.7 0.009

75 14.1 7.3 -0.001

22.9 9A -0.016

125 31.0 10.0 -0.013 150 43.0 13.0 -0.004

dimensions

perpendicular

to the director suffers the least from the constraint. A _more

isotropic

conformation is far less

probable.

The essentail feature is that back

folding

in the direction

perpendicular

to the director is

strongly hampered.

In the

model,

^the influence of the excluded- volume effect in the direction

parallel

to the director is nullified due to the

strongly

enhanced excluded-volume effect in the directions

perpendicular

to the director. For two dimensions, the situation is identical to that found for directed

SAW's,

and the SAW here is

apparently

in the

same

universality

class

[18, 19].

^Another

quantity

of interest is the order parameter s defined

aS

3

(Njj)

I

i~r d 3 ~~~

~ _"

ii

₂

~ = ~

(Ni)

~~~ ~

~ ~ j~~

(N)

where Njj ^{is the number of} steps

parallel

to the director. Despite ^the

global anisotropy

_as indicated

by

the

scaling

behaviour discussed

above,

and the

corresponding

flat

shape

_no local

ordering

is observed

(s

₌

0,

Tab.

II).

The

foregoing

results show that _as

long

_as the solvent entropy ^is

ignored,

the nematic constraint favours _a disk-like

shape

with the

longest

dimension

perpendicular

to the director.

This result

suggest

a behaviour that is counter intuitive.

Now,

the

analysis

_so far

clearly

shows

why backfolding

in _a direction

perpendicular

to the director is

hampered. However,

_as far _as the

application

to real systems is concerned _an

important

^factor

favouring

^the exact

opposite

behaviour is

missing.

If _a segment ^of the SAW is present, ^the

degeneration

^{of the solvent}

configuration

_on the line

parallel

to the director

containing

this

segment disappears,

I.e. the

configuration

is

uniquely

defined _now

(Fig. 3).

^If

only

these

entropic

losses of the solvent _are

considered,

^disk-like conformations become

extremely

unfavourable.

Hence,

ⁱⁿ

reality

we are

dealing

with at least two

opposing

effects and the _outcome is unclear.

Since,

both effects amount to a reduction in the overall entropy it is not

surprisingly

to find total

immiscibility

of flexible

polymers

in nematic low molecular

weight liquid crystals

in the limit of

long

chain

lengths [32].

j j ^{i i}

i i

i i i

i i

a ~

Fig. 3.

a)

Illustration of the twofold degeneracy of the solvent configuration _{on a} line parallel to the director in the absence of _a polymer segment.

b)

Iflustration of the uniquely defined solvent

configuration _{on a} line parallel to the director in the _presence of _a polymer segment.

(e-e)

chain segment,

(o-o)

rod.

Several

experimental techniques

like

NMR,

small

angle X-ray scattering (SAXS),

SANS

or

viscosity

measurements have been used to _measure conformational

properties

of flexible

polymers

in _a nematic medium. Our results could be relevant for very dilute solutions of

flexible

polymers

in _a nematic solvent at temperatures

sufficiently

far below the nematic-

isotropic

transition temperature, with

only

_a

slight

interaction between

polymer

and solvent.

608 JOURNAL DE PHYSIQUE II N°5

This situation _was denoted the "weak

coupling regime" by

Brochard

[20, 21],

ⁱⁿ contrast with the

"strong coupling regime",

which is characterized

by

_a local destruction of the nematic order

by

the chains. The latter situation

prevails

_near the

nematic-isotropic

transition temperature of the mixture. An

example

of the former kind of system ^is a dilute solution of

polystyrene

in the nematic solvent

N-(p-ethoxy-benzylidene)-p-butylaniline (EBBA).

^The

global

chain

shape

and the local-order parameter of this solution _were determined

by

acoustic measurements and

NMR, respectively.

It _was found that the

hydrodynamic

radius of the chain

perpendicular

to the director of the solution increases

strongly

when the temperature decreases while _on the other hand the local order parameter ^{of the}

polymer

is _very small and shows

only

_{a very}

slight

increase. The authors

actually concluded,

that if the weak

coupling regime applies,

the

polymer

coils would go ^to a flat disc _as the temperature decreases

[2-4].

A

different,

but in certain aspects

comparable,

situation arises for flexible

polymers

with

mesogenic

_groups

joined

to the

polymer

backbone

by

_a flexible spacer. The backbone shows similar

global

and local

ordering

ⁱⁿ a nematic environment created

by

the side groups, when the

length

of the flexible _spacer between the backbone and the

mesogenic

_group is not too short.

Then the conformations of the

polymer

^backbone _are

supposed

to be _{more or} less

independent

from the side groups [22]. ^SANS

experiments

_on

polymetacrylates

with

mesogenic

units

jointed

to a deuterium labeled backbone

by

_an

alkyl

_spacer, showed that the backbone _assumes

a flat conformation in the nematic melt

(S ^[

^>

S(),

^{and that} moreover this difference increases with

a decrease in temperature

[23-27].

^This is not

suprising,

since the order of the

mesogenic

groups

probably

increases with

a decrease in temperature, thus

enhanching

the flatness of the backbone conformation.

Concluding

remarks.

The _average

shape

of flexible

polymers

in _a nematic solvent is the result of _a delicate balance between

anisotropic enthapic

and

entropic

effects. The simulations show that the

anisotropic

excluded-volume effect favours _a flat disc-like

shape

of the SAW. The introduction of the flexible

polymer

in the nematic solvent reduces the entropy of the solvent. With the present model this cannot be taken into account

accurately,

since most conformations that

satisfy

the

"nematic constraint" have _a disk-like

shape

and therefore _an

extremely

small

weight

factor in the

partition

function due to the

corresponding large

reduction in solvent entropy.

Still,

our results suggest that local destruction of the nematic order

by

the

polymer

is not the

only possible explanation

for the absence of

anisotropy

in systems like

polystyrene

in

p-azoxyanisole

[2].

If, however,

favourable

enthalpic

interactions exist between the nematic solvent and the

polymer

segments

(as

for _a main-chain nematic

polymer

in

p-azoxyanisole)

the

shape

of the coil is

likely

to be stretched

along

^the director below the

nematic-isotropic

transition

point

_[6,

28-31].

References

ill

_van Wet J.H. and ten Brinke G., J. Chem. Phys. 93

(1990)

1436.

[2] ^Dubault A., Ober R., Veyss14 M. and Cabane B., J. Phys. France 46

(1985)

122?.

[3] ^Dubault A.,

Casagrande

C. and Veyss16 M., Mol. Cryst. Liq. Cryst. Lett. 41

(19?8)

239.

[4] Martinoty P., Dubault A., Casagrande C. and Veyssid M., J. Phys. Lett. Fral~ce 44

(1983)

L935.

[5] ^Brochard F., J. Polym. Sci., Polym. Phys. Ed. 17

(19?9)

136?.

[6] ^{Mattousi H. and} Veyssid M., J. Phys. France so

(1989)

99.

[?] ^{de Gennes} P-G-, Polymer Liquid Crystals, A. Ciferri, W-R-

Krigbaum

and R-B- Meyer, Eds.

(Academic,

New York,

1982).

[8] Ben-Shaul A., Rabin Y. and Gelbart W-M-, J. Chem. Phys. 78

(1983)

4303.

[9] Ten Bosch A., Maissa P. and Sizou P., J. Chem. Phys. 79

(1983)

3462.

[10] Warner M., Gunn J-M-F- and Baumgirtner A-B-, J. Phys. A: Math. Gen. 18

(1985)

300?.

[11] Wang X-J- and Warner M., J. Phys. A: Math. Gen. 19

(1986)

2215.

[12]Wang X-J- and Warner M., J. Phys. A: Math. Gen. 20

(198?)

?13.

[13] Seitz R., J. Chem. Phys. 88

(1988)

4029.

[14] ^Photinos D.J., Samulski E-T- and Toriumi H., J. Phys. Chem. 94

(1990)

4688.

[15] Baumgirtner A-B- and Moon M., Europhys. Lett. 9

(1989)

203.

[16] ^Dimarzio E-A-, Phys. Rev. Lett. 64

(1990)

2?91.

[1?] Madras N. and Sokal A-D-, J. Stat. Phys. so

(1988)

109.

[18] ^Nadal J-P-, Derrida B. and Vannimenus J., J. Phys. France 43

(1982)

1561.

[19] ^{Redner S. and} Majid I., J. Phys. A: Math. Geu. 16

(1983)

L30?.

[20] Kronberg B., Bassigana I. and Patterson D., J. Phys. Chem. 82

(19?8)

1?14.

[21] ^Brochard F., C-R- llebd. Sean. Acad. Sci. 290B

(1980)

485.

[22] PIat4 N-A- and Shibaev V-P-, Comb-shaped Polymers and Liquid Crystals

(Plenum,

New York,

198?).

[23] ^Keller P., Carvalho B., Cotton J-P-, Lambert M., Moussa F. and P4py G., J. Phys. Lett. Fral~ce L1065

(1985)

46.

[24] ^{Kirste R-G- and Ohm} H-G-, Makromol. Chem., Rapid Commuu. 6

(1985)

1?9.

[25] ^Moussa F., Cotton J-P-, Hardouin F., Keller P., Lambert M., P4py G., Mauzac M. and Richard H., J. Phys. France 48

(198?)

10?9.

[26] ^Hardouin F., Noirez L., Keller P., Lambert M., Moussa F. and P4py G., Mol. Cryst. Liq. Cryst.

155

(1988)

389.

[2?] ^Noirez L., Cotton J-P-, Hardouin F., Keller P., Moussa F., P4py G. and Strazielle C., Macro- molecules 21

(1988)

2891.

[28] ^{Volino F. and Blumstein} R-B-, Mol. Cryst. Liq. Cryst. i13

(1984)

14?.

[29] ^Volino F., Gauthier M-M-, Giroud-Godquin A-M- and Blumstein R-B-, Macromolecules 18

(1985)

2620.

[30] ^d'Allest J-F-, Sixou P., Blumstein A., Blumstein R-B-, Teixeira J. and Noirez L., Mol. Cryst.

Liq. Cryst. 155

(1988)

581.

[31] d'Allest J-F-, Maissa P., Ten Bosch A., Blumstein A., Blumstein R-B-, Teixeira J. and Noirez L., Phys. Rev. Lett.

61(1988)

2562.

[32] ^Baullaufl M., Her. Buuseuges. Phys. Chem. 90

(1986)1053.