Static scattering from cyclic copolymers in solution

(1)

HAL Id: jpa-00247881

https://hal.archives-ouvertes.fr/jpa-00247881

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.

Static scattering from cyclic copolymers in solution

M. Benmouna, R. Borsali, H. Benoît

To cite this version:

M. Benmouna, R. Borsali, H. Benoît. Static scattering from cyclic copolymers in solution. Journal de

Physique II, EDP Sciences, 1993, 3 (7), pp.1041-1047. �10.1051/jp2:1993180�. �jpa-00247881�

(2)

Classification Physi<.s Absn.acts

36.20 61.25H 5.70

Static scattering from cyclic copolymers in solution

M. Benmouna

(I, **),

R. Borsali

(I, *)

and H. Benoit

(2)

(')

CERMAV-CNRS and

University Joseph

Fourier, P-O- Box 53x, 38041 Grenoble Cedex,

France

(2) Institut Charles Sadron, 6 _rue

Boussingault,

67083

Strasbourg

Cedex, France (Received 7 Noi>ember J992, revised 29 Januaiy J993, accepted 7 Apiil J993)

Rksumd. Dans cet article, nous dtudions [es

propridtds statiques

de diffusion des

copolymbres cycliques

dans _une solution satisfaisant la condition thdta

optique.

Le _cas d'un dibloc

symdtrique

est examind _en ddtail. Les rdsultats _montrent des diffdrences

imponantes

_par _rapport _au cas d'un

copolymbre

lindaire. L'interaction

entropique qui apparait

dans _une chaine

cyclique

induit _une diminution de l'intensitd diffusde _et _un ddplacement ^de son maximum _vers _une valeur de q

plus

dlevde. Ces rdsultats montrent _un effet

compatibilisant

substantiel quand on les compare

avec ceux d'un

copolymbre

lindaire.

Abstract. The static

scattering properties

of cyclic diblock

copolymers

in solution _are studied.

The _case of _a symmetric ^diblock ⁱⁿ a solvent

satisfying

the optical theta condition is considered in details. The results show substantial differences with respect to linear diblock copolymers. The

additional entropic interaction in cyclic chains induces

a decrease of the scattering intensity ^and a

shift of the location of its maximum to a higher _q value. This results in _a substantial

compatibility

enhancement _as compared to linear copolymers.

t. Introduction.

Cyclic polymers

such _as the DNA

ring

I _are

important

because of their

potential applications

and their role in the

organization

of

living

cells.

They

are also of fundamental interest because

they provide

_a

simple

macromolecular model that can

help

in

elucidating

the basic concepts ^of the

physical properties

of macromolecular

compounds.

However, _some

special

behaviors _are

found in

cyclic

chains _as

compared

to their linear counterparts. ^In terms of their

thermodynamic properties

for

example,

measurements on dilute solutions of

polystyrene

in

cyclohexane

at 35 °C which is the theta temperature ^for ^linear

chains,

have revealed that the second virial

coefficient is

positive

for

cyclic polymers [2-4J.

This _means that the theta temperature ^is

(*) To whom correspondence should be addressed.

(**) Per.manent Addiess

University

of Tlemcen,

Physics

Department ^P-O- ^Box ^l19 Tlemcen, Algeria.

(3)

1042 JOURNAL DE PHYSIQUE II N° 7

shifted _to lower values and that the excluded volume interaction is _more

significant

than for

linear chains.

Moreover,

~ince the _monomer

density

is

higher

inside _a closed chain _as

compared

to a linear chain, _one _may _expect to see some

similarity

with branched

polymers.

Indeed, experiments reported by

the

Strasbourg

_group

[5]

on star molecules have also shown that the theta temperatures ^for ^both

vanishing

second virial coefficient and

unperturbed

dimensions _are below the theta temperature ^measured ^for the

corresponding

linear chains. The structural

properties

also show _some differences

essentially

because of the end effects which

are irrelevant when the two chain ends _are

chemically

linked

together.

It has been shown

using

static _neutron

scattering [4]

in dilute solutions of

ring polystyrene

in

cyclohexane

that the form factor _can be

represented by

the so-called Cassasa's function

P~(q) [6]

~ $

Pr(q)

=

/

^e~ ~""' d~; e'"

(i

, i~

o

u ₌

q2

N

«

~/6,

where N is the

degree

^of

polymerization,

_« the

length

^{of the} statistical segment, and _q the

scattering

wavevector q =

(4

«IA ) sin (0/2 ), A is the

wavelength

of the incident radiation and 0, the

scattering angle.

Neutron

scattering experiments

made in dilute solutions of linear chains under the _same conditions reveal that the form factor satisfies the known

Debye

function

Pt(q)= (e~"+u-1). (2)

u

The radii of

gyration

extracted from these data _are found _to

satisfy

the

relationship

R~y =

,I

R~~

(3)

where the

subscripts ^I

and

r refer _to linear and

ring polymers, respectively. Furthermore,

the

theory

of linear diblock

copolymers

in the melt has been put ^forward

by

Leibler

[7]

in the framework of the random

phase approximation (RPA) [8].

The scattered

intensity

J

(q)

in the

case of _a diblock

copolymer

is found _as follows _:

(a b)~

~

Pt(q)

~~~~ ~f~~' ~f~~l~fi(q)Pt2(q) Pij2(q)J

^~ ^~ ^~~~

a and b represent ^the

scattering lengths

of both

species

if the _neutron

scattering technique

is

used, (a b)

is the increment of the refractive index in the _case of

light scattering.

The total form factor of the chain P

(q)

_can be

expressed

in _terms of its usual internal form factors P ii

(q), Pt~(q),

and

Ptj~(q) by

the

geometrical relationship [9]

Pi(ql

=

f~Pi)(q)

₊

(i -fl~Pi~(ql

₊

2f(' -flPii~(q).

(5)

The

subscripts

I and 2 refer to the blocks A and B

respectively f

is the

composition

^of

monomer A per chain. In the case of _a

symmetric

diblock

copolymer, f=1/2,

P _tj

(q)

=

P

t~(q)

=

P

tj~~(q),

2

Pi (q)

₌ P

tj~~(q

+ P

tj~(q),

and

equation (4)

reduces to :

~a

~~~ ^~

~~ ^~ N

IF

_t

in

(q Pi (q)j

~ _x

(6)

The

plot

of

I(q)

_versus _q shows _a maximum whose

position

is located

approximately

at

q~t =

2/R~t.

^The

^height

^of ^this ^maximum

^depends

on the value of the

Flory [10]

interaction parameter X. When this parameter ^reaches a critical value _Xmt the system

undergoes

a

(4)

microphase separation

transition

(MST).

Within the RPA, the critical parameter Xm is found

as :

x~t =

~

m

~~

(7)

IN (Pmii/2 Pmt)]

^N

where the

subscript

m indicates the value of the form factor _at q = q~. The purpose of this article is to see whether similar observations _can be made for

cyclic

diblock

copolymers.

In

addition,

_we introduce _a third component ^which ^is a low molecular

weight

solvent and

investigate

the effects of

polymer

concentration in the

vicinity

of the

overlap

threshold

c ^* and above. This allows for the extension of the RPA _to solution

properties assuming

that the effects of fluctuations _can be

ignored.

2. The

cyclic copolymer

ⁱⁿ solution.

The static

scattering properties

^of ^linear ^diblock

copolymers

in solution were derived within the framework of the RPA

[I

and the results _were found _to be in reasonable agreement ^with the neutron

scattering

data

[I1, 12].

The

general

formalism is

already

known and there is _no need _to

reproduce

it here.

However,

the _case of

cyclic copolymers

has received

only

limited

attention in the literature. To _our

knowledge,

the

study

of the

scattering properties during

transesterification reactions

by

Benoit et al.

[13]

is the

only place

where the _case of

cyclic copolymers

is invoked.

Here,

_we show how the results for linear diblocks _can be extended _to closed chains. For

simplicity,

_we consider the _case of _a

symmetric

Gaussian diblock for which the

scattering intensity I(q)

is

given by [12]

1(q)

_a b 2

lPri/2(q) Pr(q)1

~

a + b

~j2 ^Pr(q)

~~ ~

l ^~ ~PN

[P~jj~(q) P~(q)]

^~ l ₊

(v

⁺

^I ^~PNP~(q)

18)

where the excluded volume parameter v can be

expressed

in terms of the

polymer-solvent

interaction

parameter

_Xo ^and ^the ^volume ^fraction ^of ^solvent

4l~

= ~P _as follows _:

v=

-2xo. (9)

1°s

For

simplicity

we have

disregarded

a

proportionality

constant which

depends

_on the molar volumes of the three components

assuming

that

they

_are

equal

to the volume of _a unit cell in the

Flory-Huggins [10]

lattice model. These numerical refinements _can be

easily

introduced whenever needed. The volume fraction of

polymer

4l should be set to I in the bulk limit when

no solvent is present. ^In ^this

limit,

_v

- ~xJ and hence

only

the first _term in the

right

hand side

(RHS)

of

equation (8)

remains and

gives

_a known result in the _case of linear diblock

copolymer

chains

[7].

The second _term in the RHS of

equation (8)

describes the scattered

intensity

_one

would obtain from _a

cyclic homopolymer having

_a form factor

P~(q),

a

degree

of

polymerization N,

and _a

slightly

enhanced excluded volume interaction because of the chemical mismatch of the _two

polymers (I,e.

_v ₊

X/2).

The essential difference between these

results and those obtained for linear chains is inherent in the form factors

P~(q)

and

P~j/~(q).

^These

quantities

_are sensitive to the chain architecture and reflect the

entropic

interactions of its _monomers. The main

assumption

made in their calculation is to write the average square distance between _two _monomers I and

j separated by

the chemical distance

JOURNAL DE PHYSIQUEII T N' 7 JUQ 199~ 4?

(5)

n =

ii j

as

[14]

(r()

= n

(1

_N^~

^«2 ⁽¹⁰⁾

where the

symbol (...)

denotes the thermal average. The second _term in the RHS of this

equation

is due _to the fact that _one _can go from _to

j

_monomers

following

two different

paths along

the chain.

Using

this

assumption

in the definition of the total form

factor,

_one obtains _:

~r(~)~ 0

dXe ^"~~ ^'~

(ii)

which

yields

the Cassasa function in

equation (I).

The calculation of P

~jj~(q)

can be made in _a similar way but the _n values should be limited _to the range from to N/2. One obtains the

following simple

result

jl

e~ ~*~~]

(12) p~~~~(q)

₌

~~@)

An

interesting _property

of

equation (8)

is that the two terms carry

physical meanings

of different natures. It shows that the

scattering

due to the local

composition (or

the concentration

difference)

fluctuations is

completely decoupled

^from the

scattering

due _to the total

concentration fluctuations of the

polymer

in the solvent.

Furthermore,

_one _can have direct

access to each of these terms

by choosing properly

^the

scattering lengths

_a, b, and

s or

equivalently

the indices of refraction

depending

on the type ^of radiation to be used.

Obviously,

the

composition

fluctuations _are _more relevant and

they

_can be evaluated

unambigously

in the so-called _zero average ^contrast condition

[12J

or

equivalently

for

symmetrical

mixtures, in the

optical

theta condition

[14J.

This _means that

and from

equation (8)

1(q)

a b

j~ ^lPrl/2(q) ^Pr(q)]

I

₂

x

~'~)

l

j

^*N

lPrli2(q) Pr(q)]

First,

_one notes that the structure factor

multiplying

^~ ^~ ⁱⁿ ^this

equation

is

proportional

2

~

to

Sy(q)

which is defined

by

:

where

S(q), S(q),

and

S~~(q)

_are the

partial

structure factors due to interferences between

AA, BB,

and AB molecules. In the

symmetrical

_case, _one has

[12]

_:

PTl12(~)

+ V*NP

T(~)iPrl/2(~) PT(~)i

~~~~~~

~

~~~~~~

~

~~~~

_~

4 D

(q)

^~~~~~

Pj2(~)

IV +

X) *NPTI~)iPTl/21~) Pri~)i sab(~)

~

sbai~)

"

s~(~)

"

~ j~~~~

i16b)

(6)

and

D(q)

=

(1-

^~₂

~PN(P~j/~(q) P~(q))j [I

⁺

(v

⁺ ^~₂

~PNP~(q)j ^(16c) Combining equations (15)

and

(16) yields

_:

Sy(q)= ~'~ ^~~~~ ₍₁₇₎

1-j#Nip~jn(q)-p~(q)j

3. Discussions and conclusions.

Figure

shows the variations of the form factors

Py(q) (curve a), P~(q) (curve b),

and P j~~

~(q) (curve c)

as a function of q. The curves a and b _are

represented by

the

Debye

and the Casassa

functions, respectively,

and _are found _to describe

quite

well the _neutron

scattering

data from dilute solutions of linear and

cyclic polystyrene

chains in

cyclohexane [4].

One observes that, at _a

given

value of q, the effect of intramolecular interferences is weaker for _a half than for _a total

ring

which in tum shows weaker interferences than _a linear chain. The latter

behavior _can be understood since the local

density

of

interfering

_monomers is

higher

in the

case of _a

ring.

In

figure2,

_we have

plotted

the normalized

scattering intensity 41(q)/[ la b)~

~PN _as _a function of q for linear

(curve a)

and

cyclic (curve b) copolymers

made of _monomers of

practically

the _same chemical nature with _zero interaction parameter

(I,e.

_X

=

0). They

differ

only by

their

scattering lengths

which

satisfy

the condition in

equation (10).

The

interesting

remark is that the normalized

intensity 41(q)/[ la b)~

~PN is

equal

to P

j~~(q) P~(q) independently

of the

polymer

concentration. Similar behavior _was observed in the _neutron data of half deuterated

polystyrene

linear chains in _a

symmetrical

mixture of

ordinary

and deuterated benzene

[12]

above the

overlap

concentration _c *. It would be

interesting

to check whether _a similar observation could be made for

cyclic copolymers.

Obviously,

this conclusion is modified when the _two blocks _are different and _a finite

parameter

o

~

o.

f

~ o.

0.

0.02 0.04 0.06 0.08 o-1 0.12 0.14

~(~-l)

Fig.

I. The variation of the forrn factors _as _a function of the _wavevector q. Curve _a _: the _case of _a linear chain

having

the radius of

gyration

_{R~t (I,e.}

Debye's

function) Curve b the _case of _a cyclic chain

having

the radius of

gyration

_R~~ (I.e. Casassa's function) Curve _c represents ^P

ji?r(q)

the form factor of _one block (see Eq. (12)).

(7)

I

_o.

4

~

i

o-1

~

i

o.05 ~

0.02 0.04 0.06 0.08 o-1 0.12 0,14

~tJK-')

Fig. 2. The variation of the normalized

intensity

_a~ _a function of the wavevector q for diblock copolymers made of

compatible

_monomers (I.e. _X 0). Curve _a _: linear symmetric diblock. Curve b

cyclic

symmetric diblock.

0.

0.4 _a

~

f

_0.

I

i

~ 0.

,

~

#

~

~t~-I )

Fig. 3. The variation of the normalized intensity _as _a function of the _wavevector _q for diblock copolymers ^made of

interacting

monomers at a value of the interaction parameter ⁱⁿ ^the vicinity of the MST for linear diblocks (x 4lN

= lo ). Curve _a _: linear

symmetric

diblock. Curve b

cyclic symmetric

diblock. The

following

parameters were used in these

plots

N ₌

lo',

and _«

= 5 h.

X characterizes their

incompatibility.

The

position

of the maximum is

practically

not modified

by

_X or ~fi but its

height

increases

substantially

with

increasing

_X _or ~fi. In

fact,

if the

quantity X~fiN

reaches the value

2/(P

j~~~

P~~ ),

the _structure factor

St (q) diverges

and the

system undergoes

an MST.

Figure

3 shows the variation of the normalized

intensity

_as a function of q for _X 4lN

=

lo which

corresponds practically

to the condition of MST for the linear chains.

One _sees that the

intensity

for

cyclic copolymers

remains finite and

comparatively

small. This

(8)

enhancement of

compatibility

is _an essential

property

^of ^mixtures

involving cyclic polymers

and _was observed

by

Cates and Deutsch

[16]

for the first time. It is

interesting

to note that for the

microphase

transition to take

place

in the

ring

system at the concentration considered in

figure

3

(and

for which linear

copolymers undergo

_an

MST)

_one must take _a critical

parameter

for MST Xmr such that _Xmr ~PN=18 which is about

eighty percent higher

than the

corresponding

value in the system of linear chains. If these results _are confirmed

by

the

experiments,

it would _mean that the present mean field model remains valid for

cyclic copolymers

much

beyond

its range of

validity

for the

corresponding

linear

copolymers.

These

observations _can be tested

by

neutron or

light scattering techniques

and _a proper choice of the mixture to

satisfy

the so-called

optical

theta condition. In neutron

scattering,

_one _can _use

diblock

copolymers

made of

ordinary

^and ^deuterated monomers in _a mixture of

ordinary

and

deuterated

good

solvents. The drawback of this system ^is ^that ^it shows _a very small _x-

parameter. ^In

light scattering

there _are several systems

fulfilling

the above condition and _some of them _were

already

studied in the _case of linear chains

[15]. Finally,

_one observes in

figures

2 and 3 that the

position

of the maximum q~ is shifted _to

higher

values

indicating

_a shorter scale of the microstructure which may appear in the _case of

ring copolymers.

Acknowledgements.

We thank the referee for

bringing

references

[2], [13],

and

[16]

_to _our attention and for

correcting

an erroneous statement on the shift of the theta temperature ^for

cyclic homopolym-

ers. M. Benmouna thanks the Fondation

Scientifique

de

Lyon (Rdgion Rhbne-Alpes)

for

financial support ^and ^Madame ^M.

Rinaudo,

head of

CERMAV,

for

hospitality.

References

[I] WEIL R., VINGRAD J., P/.oc. Natl. Acad. St-I. (USA) So (1965) 730.

[2] al ROOVERS J., ToPoRowsKi P. M., Macromolecules 16 (1983) 843 _; b) RoovERs J., Mac.iomole<.vies 18 (1985) 1359.

[3] ZIMM B. H., ^STOCKMAYER ^W. H., J. Chem. Phys. 17 (1946) 130.

[4] HADzIIOANNOU G., Cows P. M., BRINKE G. ten, HAN C. C., LUTz P., STRAzIELLE C., REMPP P., KovAcs A. J., Macromolecules 20 (1987) 493, and references cited herein.

[5] CANDAU F., REMP P., BENOIT H., Macromolecules 5 (1972) 627.

[6] CASASSA E. F., J. Pot>,m. Sci., Pal-t A 3 (1965) 605.

[7] LEIBLER L., Macromolecules13 (1980) 1602.

[8] DE GENNES P. G.,

Scaling Concepts

in

Polymer Physics

(Cornell

University

Press, Ithaca, 1979).

[9] IoNEsco L., PicoT C., DUPLESSIX R., DUVAL M., BENOIT H., LINGELSER J. P., GALLOT Y., J.

Polym. St-I. Polym. Ph_vs. Ed. 19 (1981) 1033.

[10] FLORY P. J..

Principles

of

Polymer Chemistry

(Cornell

University

Press, Ithaca, 1953).

[I Ii BENOfT H., Wu H., BENMOUNA M., BAUER B., MOzER M., LAPP A., Macromolecules 20 (1987) 381.

[12] DUVAL M., PicoT C., BENMOUNA M., BENOfT H., J. Phy.I. Fran<.e 49 (1988) 1963.

[13]

BENOIT H., FISCHER E. W., ZACHMANN H. G.,

Polyniei

30 (1989) 379.

[14] YAMAKAWA H., Modern Theory ^of Polymer ^Solutions (Harper ^and Row, New York, 1971).

ii 5] OULD-KADDOUR L., STRAzIELLE C., Polymer 28 (1987) 459 and references cited herein.

[16] CATES M. E., DEUTSCH J. M., J. Phys. Fiance 47 (1986) 2121.