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To cite this version:
B. Farnoux, M. Daoud, D. Decker, G. Jannink, R. Ober. Shape change of the flexible polymer in
solution with increasing concentration. Journal de Physique Lettres, Edp sciences, 1975, 36 (2),
pp.35-39. �10.1051/jphyslet:0197500360203500�. �jpa-00231148�
L-35
SHAPE CHANGE OF THE FLEXIBLE POLYMER
IN SOLUTION WITH INCREASING CONCENTRATION
B.
FARNOUX,
M.DAOUD,
D.DECKER (*),
G. JANNINK and R.OBER(**)
Service de
Physique
du Solide et de RésonanceMagnétique
Centre d’Etudes Nucléaires de
Saclay,
BP2, 91190 Gif-sur-Yvette,
FranceRésumé. 2014 Une expérience de diffusion de neutrons par des mélanges de polystyrènes deutériés
et protonés en solution dans le sulfure de carbone a permis d’observer l’évolution de la configuration
de la chaine en fonction de la concentration. La dépendance en concentration de la fonction de corrélation de paire de segments appartenant à une même chaine est décrite à l’aide de l’exposant
de volume exclu.
Abstract. 2014 A few deuterated polystyrene chains are
dispersed
in a solution of protonated polystyrene and carbone disulfide. The neutron scattered intensity is measured at several concen-trations of protonated chains.
Already
in the semidilute range, there is astriking
contrast withthe results obtained at equal concentrations in
fully
deuterated solutions. The dependence of the pair correlation function on concentration is discussed in terms of the excluded volume exponent.The excluded volume
exponent v
has been mea- sured[1 ]
in dilute solutions(v ~ 3/5)
and in the bulk state(v
=1/2),
and thequestion
arises as to whathappens in
the intermediate situation when thepolymer concentration
C varies from zeroto d,
thepolymer density.
Such aproblem
istypical
of a situa-tion in which there is a cross over
region
for the critical exponents. The parameter is here the concentration C.The
experimental
answer can begiven by
the obser-vation of the
single
chain in solutions withincreasing polymer
concentration. Thisrequirement
isnearly ideally
achievedexperimentally by observing
thescattering
of neutrons from mixtures of deuterated andprotonated
chainsdispersed
in asolvant,
suchthat the contrast between
protonated
chains and solvant is very smallcompared
to the contrast betweendeuterated chains and solvant and can be
neglected.
The neutron
intensity
scatteredby
deuteratedpolystyrene (concentration CD) dispersed
in solutionsof
protonated polystyrene (concentration Cp)
andcarbon disulfide was therefore measured at different values of
CD
andCp (Table I).
In apreliminary
expe- riment thescattering
law was tested forCp
= 0 andCD
>C*,
the concentration at which the chainsbegin
tooverlap.
For thisexperiment,
thescattering amplitude
is written as~f(~), where q
is the momen-tum transfer. In the main
experiments,
thescattering
law was measured for
CD
C* andC = Cp + CD >
C* .(*) CRM, 6, rue Boussingault, Strasbourg.
(**) College de France, 11, place Marcellin-Berthelot, Paris.
Here, analogous
to theabove, S~ (q),
denotes thescattering amplitude.
Theexperiments
were carriedout in the intermediate momentum range
where I is the step
length
and N is the number of steps per chain. Earlier results[2, 3] showed
that inrange
(1)
the theoreticalexpression
was verified where A is a constant
and ~
is the screen-ing length
in which v is the excluded volume
parameter, A
theAvogadro number,
and m is the molecularweight
of a
segment.
Form(2)
will serve as a reference in theinterpretation
of theexperiments,
for which theanalytical
form5~)
is not known. In fact~(~)
isknown in the two limits C -~ 0 and C =
d,
where it isHere v takes the value
3/5
for C -~0,
and1/2
forC = d. B is a constant. An
expression
similar toeq.
(4)
isexpected
to be valid in the above mentionedArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:0197500360203500
0) Values obtained from the table VI of macromolecules (to appear in 1974).
(b) The molecular mass of the protonated polystyrene are the same than the deuterated one.
concentration range. However the momentum range for which the exponent v takes a definite value may not cover the entire range
of q
values(1).
Theproblem
is to determine the momentum
dependence
of~(~)
for C * C d from the
experimental
datareported
in this letter
(Fig. 1).
These data were obtained with the smallangle scattering apparatus
at the neutronguide
exit of reactor EL3 atSaclay.
Thewave-length
of the incident
beam,
selectedby
acrystalline
mono-chromator was À = 4.62 ± 0.04
A,
and adivergence
of less than 0.3
degree was
achievedby
means of acollimator. The scattered
intensity
was recordedfor
angles
0 so chosen that the momentum transfer q =4 ~/~,
sin0/2 ranged
from10 - 2
to10 -1 A-1, satisfying inequality (1).
Theintensity
scatteredby
the
labelled
chains was obtainedby substracting
theintensity
scatteredby
the solvant(i.e.
carbon disulfideplus protonated chains)
from the totalintensity.
Astudy
of themultiple scattering
effects made for aprevious experiment [1 ],
had shown that these did notinterfer with the results.
Before
discussing
theexperimentally
observedresults we consider
representations
of the scatteredintensity
versus momentum transfer which are appro-priate
to determine the values of v in interval(1).
The
interpolation
formula willessentially
be eq.(4),
in which
however,
abroadening
D due to the finiteFIG. 1. - Scattered intensity as a function of momentum transfer q.
Sample F. The lower curve (closed circles) is the background due
to solvent, container and incoherent scattering of the solute mole- cules. This curve is subtracted from the upper one to give the useful signal. The momentum range is divided into two parts corresponding
to appropriate representations of the results. (See text.)
L-37
concentration
CD
of deuteratedchains,
is allowed to appearA )
In thelimit q , (N 1/2 l ) -1,
the most appro-priate representation
of theexperimental
data is(S~ (q))-1
versusq2
= X. . In case the deuteratedchains behave as isolated self
avoiding
chains in agood solvent,
then(sc (R’)) 1
=X516,
X >(Nl)-2. (5)
The curvature in
(5)
is the morepronounced
thenearer X is allowed to
approach
zero. In the limit ofan infinite
polymer chain,
curve(5) extrapolates
to avertical tangent.
If,
on the contrary, the deuterated chains behave asfree
coils, (S~ (q))’~ 1 depends linearly
on X.A similar
representation
consists inplotting
theresults
(5~))~
versusq5~3
= Y. The free coil model isrepresented by
(S.L(q))- I
=Y6~5~
Y >(Nl ) - $~3 (6)
whereas the chain with self excluded volume inter- action
yields
(S,L(q))- 1 = Y,
Y >(Nl)-~~3 . (6’) B )
In the upperpart
of range(1),
a convenientrepresentation
of theexperimental
data isS~ (q)
versus Z =
1/~.
Moreprecisely,
If the
exponent
v isstrictly greater
than1/2 (for
instance v =
3/5),
curve(7) displays
a curvature.On the other
hand,
if v =1/2
this function is linearprovided
that{ D
+O(IIN111~’) I
is smallenough,
i.e. smaller than
Zm~.
C)
Thelog 5~)
versuslog q representation
usedin reference
[ 1 ]
is notappropriate
for theinvestigation
of the lower part of range
(1)
because of the broaden-ing
termsç-2
andD, respectively,
in(2)
and in(4’).
In the upper part of range
(1),
thisrepresentation
is asgood
asB)
andyields
identical results.With these considerations in
mind,
we shall deter-mine,
from examination of the scatteredintensity
curves, which of the two values of the exponent v
provides
the best fit with thedata,
individing perhaps arbitrarily
momentum range(1)
into two intervals.For the semi-dilute
solutions,
we shall observe thelimit q -+ (Nl ) -1,
since it isexpected
that the concen-tration effects first
modify
the lowerpart
of(1),
as Cincreases. In
figures
2 and3, (S~ (q))-1
and(S~ (q))-1
~ . ~ -
FIG. 2. - Inverse scattering laws
(S~ (q))’
1 (closed circles) and(S~ (q)) -I
(crossed circles) as a function of a)q2
and of b) qS/3,for C = 4 x 10-2
g.cm-3.
Also represented is the scattering bya dilute solution
(Scf(q))-l (open
circles).I FIG. 3. - Inverse scattering laws
(S~ (q))-1
(closed circles) and(S~ (q))-1
(crossed circles) as a function of a)q2
and of b)qs~3,
forC=7.5xlO’~g.cm’~.
the
scattering So by
a very dilute solution(sample A).
The
opposition
betweenS’[
andS~
is obvious and the curvature ofSeL
appearsclearly
infigs. (2a)
and(3a).
On the contrary,
S~
and5’J
show identical behaviours.At these two finite concentrations
(C
>C*),
themomentum range over which v =
3/5
holds truecovers most of range
(1),
as in the case of very dilute solutions(sample A).
At the
higher
concentrationC = 50 x 10- Z g . cm- 3,
the
picture
is different(Fig. 4a).
The inversescattering
law
(S~ ) -1
as a function ofq2
does not indicate the curvature observed in the semi-dilute solutions (Fies. 2a and 3a).FIG. 4. - Inverse scattering law
(S~ (q))-1
a) as a function ofq2,
b) as a function ofq5J3,
for C = 50 x 10-2g.cm-3.
In b) a tangentis drawn to help visualize the curvature.
in the
limit q , (Nl) - 1,
the excluded volume expo- nent for thesingle
chain takes the value v =1/2.
This result
encouraged
us toinvestigate carefully
theupper part of momentum range
(1).
Infigure 5,
thescattering
law for C = 50 x10- 2 g . cm- 3 (5a)
andC = d
(5b)
isdisplayed
as a function of1/~.
Thebulk
picture
serves here as areference,
since it isalready
known from earlierexperiments
that the valuev =
1 /2
holds true over the entire range(1).
This isnot the case at C = 50 x
10- 2 g . cm- 3
which dis-plays
a curvature in the1 /q~ representation.
Thiscurvature cannot be attributed to the term D in
(7) ; namely
thehighest possible
value of D which is the inversesquared screening length
associated with apolymer
concentrationequal
toCD (D ~ 10-4 A-2).
The curvature is in fact due to the value of the excluded volume exponent v =
3/5.
FIG. 5. - Scattering law
~(~)
as a function of1/~
fora) C = 50 x 10-2 g.cm-3
and b) C = d.The earlier
picture given
in reference[1] may
nowbe
completed
in thefollowing
way.1)
Semi-dilute solutions. - The effect of the uniform distribution of the chain segments in the solution is borne outby
the Lorentzian behaviour of the total.
scattering
law5’~) (i.e.
all segmentscoherently scatter).
In contrast, thescattering
law for thesingle
chain
S~ (q) (i.e.
a few chains aredeuterated)
stillL-39
reflects the
self
excluded volume effect v =3/5
overmost of the intermediate momentum transfer range
( 1 ).
2)
Concentrated solutions. - The behaviour ofS,,’
in the lower part of the momentum range
departs
from the isolated chain model. The exponent
v =
1 /2
fits the data in thisregion.
On thecontrary,
in the upper part of the momentum range thechain
still behaves as if its conformation were dominated
by
its own excluded volume. The
exponent
is herev =
3/5.
Such distinct behaviours lead to a cross-overmomentum value
q*.
This momentum increaseswith concentration. It is however different from zero
only
for concentrationslarger
than 10 x10 - 2 g . cm - 3 .
3)
In the bulk state v =1/2 uniformly
over theentire intermediate momentum range. The momen- tum
q*
has increasedbeyond l -1.
An intuitive
interpretation
of result(2)
may begiven by
examination of the force field[4]
aroundthe labelled
polymer segment.
This fieldV(r)
is pro-portional
to the segmentdensity.
In the limit C =0,
this isV(r)
=kTvl - 2 p(r)
where
p(r)
is thedensity
of segments for a chain whose firstsegment
is fixed at theorigin. (p(r)
isequal
to the pair correlationfunction.)
As the concen-tration
increases,
the segment at theorigin
is alsosurrounded
by
otherchains,
whosesegment density
is
represented by
the horizontal line p infigure
6.At the intersection
r*,
the two densities areequal.
For distances r >
r*,
these f interacting potential
is
strongly
screenedby
the presence of otherchains ;
whereas for r
r*,
theself excluded
volume effect still dominates. Asimple
derivation shows thatr* _
(q*) ~
1/2
,* =
(q*)-l = ç( tYl2 ,
which is not a very
satisfying scaling
law from atheoretical
point
ofview,
nor does it fit thepresent
observations.However,
it satisfies theinequality
I r* R
FIG. 6. - Schematic view of the segment densities surrounding
the first segment of a chain in the concentrated solution. a) density
of segments belonging to its own chain, b) density of segments belonging to other chains.
and
gives
anapproximate
idea of the cross over line.This idea contradicts the
opinion
that the index v decreases from the value3/5
to the value1/2
as afunction of
increasing
concentration. At agiven (high)
concentration there is a value q =q*
whichseparates
thereciprocal
space into two domains.For q
q*,
the chain has afree
coil behaviour(v
=1/2) ;
for q >q*,
the chain has an excluded volume behaviour(v
=3/5).
This value ofq*
movesfrom
R -1
to1 - 1
as the concentration increases from the dilute to the bulk state.Acknowledgments.
- We wish to thank Prof.P. G. de Gennes for
suggesting
anddiscussing
thisexperiment,
Prof. H. Benoit and Dr. J. P. Cotton for manyhelpful
criticisms of themanuscript.
References [1] COTTON, J. P., DECKER, D., FARNOUX, P., JANNINK, G., OBER, R.,
PICOT, C., Phys. Rev. Lett. 32 (1974) 1170.
[2] COTTON, J. P., FARNOUX, B., JANNINK, G., J. Chem. Phys.
57 (1972) 290.
[3] COTTON, J. P., Thesis Paris 1973, C.E.A-N-1743.
[4] DE GENNES, P. G., Rep. Prog. Phys. 34 (1969) 1253.
[5] JANNINK, G., DE GENNES, P. G., J. Chem. Phys. 48 (1968) 2260.