Interpretation of small angle neutron scattering by copolymer solutions as a function of concentration

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Interpretation of small angle neutron scattering by copolymer solutions as a function of concentration

M. Duval, C. Picot, M. Benmouna, H. Benoit

To cite this version:

M. Duval, C. Picot, M. Benmouna, H. Benoit. Interpretation of small angle neutron scattering by copolymer solutions as a function of concentration. Journal de Physique, 1988, 49 (11), pp.1963-1968.

�10.1051/jphys:0198800490110196300�. �jpa-00210875�

Interpretation ^{of small} angle ^neutron scattering by copolymer ^solutions

as a

function of concentration

M. Duval, ^C. Picot, ^M. Benmouna

(*)

and H. Benoît

Institut Charles Sadron, (CRM-EAHP)-(CNRS-ULP), ^{6 rue} Boussingault, ⁶⁷⁰⁸³ Strasbourg Cedex, ^France (Reçu ^{le 13} juin 1988, accepté ^{le 27} juillet 1988)

Résumé. ²⁰¹⁴ Nous avons utilisé une généralisation ^de l’ Approximation de Phase Aléatoire (RPA) appliquée à

un système ^{à trois} composants pour calculer l’intensité diffusée par ^un copolymère à deux blocs dans deux cas

particuliers ^où: a) ^la longueur ^de diffusion cohérente du solvant est égale ^{à la} longueur ^de diffusion cohérente de l’un des blocs ; b) la différence entre la longueur ^de diffusion moyenne de l’ensemble de la molécule ^et celle du solvant est nulle. Nous donnons les expressions ^{formelles et} les discutons en fonction de la concentration et du vecteur de transfert q ^en supposant que le solvant ^est bon pour les deux séquences. Il y ^{a un} bon accord entre la théorie et les résultats expérimentaux obtenus par diffusion de ^{neutron aux} petits angles ^{sur un} copolymère biséquencé ^PSH-PSD (Polystyrène hydrogéné ^et deutéré) ^en solution dans un mélange ^benzène hydrogéné/benzène ^deutéré.

Abstract. ²⁰¹⁴ Using ^a generalization ^{of the} Random Phase Approximation (RPA) ^{for a} three-component system, ^the intensity ^scattered by ^{a two-block} copolymer has been calculated in the two special ^{cases :} a) ^the

coherent scattering length of the solvent is identical to the coherent scattering length ^{of one of the blocks,} b) ^the difference between the average scattering length of the whole molecule and the solvent is zero.

Expressions ^are explicitly given and discussed as a function of the concentration and the momentum transfert q assuming the solvent to be good for both sequences. Experimental results obtained by ^small angle ^neutron scattering ^{on a two block} copolymer ^PSH-PSD (hydrogenated and deuterated Polystyrene) ^{in a mixture of} hydrogenated and deuterated benzene show a good agreement between the theory ^{and the} experiments.

Classification Physics ^Abstracts

61.12 ^- 78.35

1. Introduction.

During ^{the recent} years many results ^{on neutron} and X ray scattering by

copolymers

in bulk have been

published.

^{It seems} that, ⁱⁿ homogeneous systems, the Random Phase

Approximation (RPA)

^describes

the

experimental

^results rather well

[1-3].

^{Since it}

has been possible ^to

generalize

the results of this

theory ^{to solutions}

[2,

^4,

5]

^{it was} interesting ^to

check whether the

approximation

^{is as}

good

^for

solutions as for the bulk. Some results have already

been

published [6]

^{but it was} interesting ^{to have}

qualitative

general ^laws describing ^{what kind of}

results can be

expected depending

^{on the structure}

of the polymer, its concentration and the ther-

modynamical interaction parameters.

In principle ^{this can} be extracted from the general

equation

given by ^{Benoit et al.}

[2]. Unfortunately,

because of its generality, it is difficult to handle due

to the large ^{number of}

adjustable

parameters. ^{In this} paper ^we have focused the theoretical developments

on the model of a two-block copolymer ^{AB in which}

both sequences obey ^{the same} statistical laws and have the same radius of gyration

RG.

^{It will be shown}

that this

approximation

^{does not limit the} generality

of the results at least for two-block copolymers.

In the experimental ^{section we shall} give ^some

results which, taking ^{into account the}

polydispersity

of the sample, ^{show that the} theoretical predictions

are confirmed below and above the overlap ^concen-

tration c*.

2. Theoretical section.

2.1 GENERAL EQUATION. ^- Let us consider a

diblock copolymer made of units of types ^{A and B in}

a solvent S. The ratio of the volume of one block to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490110196300

1964

the volume of a solvent molecule is defined as

Na

⁼

Va/V, and Nb

⁼

Vb/VS.

Calling n ^{the number}

of copolymer ^molecules per unit volume, ^{the volume}

fraction cp of the polymer ^{is written as :}

where N ⁼

Na

⁺

Nb and NS

^{is the number of solvent}

molecules per unit volume. The

quantity

^{Nn +}

NS

^{can be} considered as the number of unit _{cells per} unit volume.

The volume fractions cp a and _{9b of} species ^{A and}

B are defined by :

with

The structure factors

P; (q)

^{of each block of the} copolymer ^{is defined} by :

where

subscripts j

^{and k refer to monomers}

pertain- ing

^{to the same block A or B} in the chain. In this relation q is the scattering ^vector

(q

⁼

(4 ?r/A )

^sin

(J /2; À being

^the wavelength ^{of the}

incident beam and 0 the scattering

angle).

^{In the}

following

expressions any mathematical form of

P (q )

^{can be used but, to} make calculations simpler,

all the figures have been drawn using ^for

P (q )

^the

Gaussian approximation. ^We define the cross struc- ture factor

Pab (q )

^{as :}

where i and j

belong

^to sequences ^of different chemical species. ^This

quantity

^{is no}

longer

purely geometrical. ^{The total structure} factor is given by

[6] :

¹

With these notations the scattered

intensity

per unit cell

I (q )

^{can be written as}

[2] :

where a, b and s ^are the coherent

scattering

lengths

per unit volume and va, Vb ^and Vab ^{are the dimension-} less excluded volume parameters

characterizing

^the

interactions A-S, B-S and A-B respectively.

2.2 APPLICATION TO SYMMETRICAL DIBLOCK COPOLYMERS. - This equation ^{is difficult to handle.}

In order to

simplify

^the discussion we shall assume

that

Pa (q )

⁼

P b (q )

^{which means that the two blocks}

have the same shape ^{and the same} dimensions and that they occupy ^{the same} volume

(Va

⁼

Vb, Na

⁼

Nb

^{and cm =} J3 ⁼

0.5).

^{Moreover it will be}

assumed that the solvent is

good

for both sequences and Va ⁼ Vb ^{= v.} With these notations and

knowing

from the relation

(6)

^that

Pab

⁼

2 PT - Pa

⁼

2

PT - P12, equation (7)

^can be written as :

This leads to introduce the factor

z (q) :

and

Generally

speaking I (q )

^{is the} average value of two terms. The first one with

z = 1 (9 = 0)

corresponds

to the dilute solution term. The second one corres-

ponds ^{to the bulk state} where it is known

[4]

^that

v-+oo and z=0.

In order to see more clearly ^{how these terms}

influence I

(q )

^{we shall now examine some}

particular

cases :

i) a = b.

As

expected,

the classical Zimm

equation

^is

recovered. The problem ^of copolymers ^{reduces to} homopolymers.

ii) b

^{= s.}

This is the case where the solvent matches one of the sequences and

equation (10)

^{reduces to :}

The first term characterizes the behaviour of

I(q)

in the dilute

regime.

The second term domi- nates in the upper concentration range. It is seen

that, ^{in the bulk} limit, the known result

[7]

(P l2 - PT )

is recovered. To illustrate this behaviour

we have plotted ⁱⁿ figure ^{1 the} variations of

4 I (q)/ (a - S)2 cpN

^{as a} function of

qRGT

^{in the}

three concentration regimes, namely the dilute

(vcpN

⁼

0.1),

the semi-dilute

(vcpN =1)

^{and the}

concentrated

(vcpN == 10)

regimes. RGT is ^{the radius}

of

gyration

of the whole copolymer molecule. One observes that a maximum appears ^as vcpN ^increases.

The existence of the maximum is related to the initial slope ^of

I (q ).

^For v cpN ^{1 the initial} slope ^is negative ^{while for} v oN > ^{1 it is} positive. Straightfor-

ward calculations show that it becomes zero for

vcpN ⁼

J2 - 1.

This value corresponds ^{to the dilute} regime. We recall that vcpN ^{= 1 defines a concen-}

tration of the order of c*.

The analysis ^{of the curves of} figure ^{1 is} extremely simple if the fact that z depends on q is ^not taken into account. The variation of

1/z

^{as a function} of q

is drawn in figure ^{2 where} it is shown that this parameter goes ^to unity regardless ^{of the} vcpN ^value

Fig. 1. - Theoretical scattered intensity by ^{a diblock} copolymer ^A-B (50/50) ^{as a} function of qRGT. ^The

sequence B has ^{no contrast} with the solvent (see Eq. (11)).

From top ^{to bottom} vgN = 0.1, 0.4, 1, ^10.

Fig. 2. - Variation of 1 /z ^{as a} function of

(QRGT )2

(see Eq. (9)) ^{at different} vcpN values ; ^from top ^{to bottom} t;pN=8,4,l,0.

when q increases. In fact, except ^{at z = 0}

(voN-+oo)

^{or z =1}

(vcpN -+ 0),

there is a q

dependence ^{on z. This} dependence ^is mostly appa- rent in the intermediate range of concentration but it remains small in the low and high concentration range.

iii)

Zero average ^contrast condition.

This condition is fulfilled when the solvent has an

intermediate scattering length

(i.e.

^{a - s =}

-

(b - s )

^{or s =}

(a

⁺

b )/2).

This limit is interesting

because it leads ^to results similar to those in the bulk. In this case

equation (10)

^{reduces to :}

This

expression

^shows that, ^{in such contrast con-} ditions, the relative intensity

I (q)/ cp

^{remains con-}

stant whatever the concentration from the bulk to dilute solution.

2.3 DISSYMMETRIC DIBLOCK COPOLYMERS. - Up

to now, it was assumed that the statistical units of both sequences have identical lengths. ^{This is a}

crude assumption ^{since one is} rarely ^{able to} prepare

exactly ^{50/50 block} copolymers. The theoretical formalism that we have at our disposal ^{enables us to}

account for any kind of dissymmetry

(radius

^of

gyration,

volume, composition, thermodynamical parameters,

etc.).

Nevertheless the introduction of these factors leads ^to tedious calculations and it is not easy ^{to extract} simple information from the final

expression.

^{In the} present ^{section we introduce the} effect of dissymmetry in size and composition ^{of the}

two blocks. In such a situation, ^the expression ^{of the}

scattered intensity ^can be written as :

This relation has the same structure as

equation (10)

with a first term

corresponding

^to the dilute

regime

and a second one which can be associated with the bulk limit. When the sequence ^{B has no contrast} with respect ^{to the} solvent,

equation (13)

^{becomes :}

which is quite comparable ^to

equation (11).

^There-

fore the effect of the concentration should not be very different from the symmetrical ^{case. For the}

1966

zero average ^contrast condition equation

(13)

^re-

duces to :

This relation reduces to

equation (12)

^{for a = 1/2.}

The effects of concentration and composition ^are

illustrated in figure ^{3 where we} have considered the

Fig. 3. - Theoretical scattered intensity by ^{a diblock} copolymer ^{A-B as a} function of qROT. Zero average contrast limit (see Eq. (15)). ^From top ^{to bottom a} = 0.1, 0.3, ^0.5. (- vípN ⁼ 0.1 ; ^---- vípN ⁼ 10.)

limits of infinitely dilute solutions and bulk. The

following ^{remarks can be made :}

i)

^The effect of concentration is weak for all

dissymmetrical

^diblock copolymers and cancels at

a = 1/2. In fact the second term of

equation (15)

^can

be

simplified

^{as :}

at low concentration at

high

concentration

which are identical expressions ^{for a 50/50 diblock} copolymer.

ii)

^The difference in shape ^{between the} peak ^at high concentrations and the peak ^{at low concen-}

trations is very small. Both ^curves

begin

^{with the}

same apparent ^{radius of} gyration

(Ra + Rb - 2 R 2

and have the same

asymptotical

^behaviour.

iii)

^{As the}

dissymmetry

increases, ^the

peak

^be-

comes broader and shifts to

higher

^values of q regardless of the concentration.

3. Experimental ^section.

3.1 SAMPLES AND SMALL ANGLE NEUTRON EXPERI- MENTS. ^- In order to check the

validity

^{of the}

theoretical results presented ^{above we have investi-} gated ^{the case of a diblock} copolymer ^PSH-PSD by

the small angle ^neutron

scattering technique (SANS).

^This

technique

allows the study ^{of the}

scattered intensity ^{in a} convenient range of q ^values

where the maxima which are expected ^{from the} theory, ^{in zero} average ^contrast conditions, ^should

be observable.

The PSH-PSD copolymer ^was prepared by ^anionic

copolymerization

following ^{a classical}

procedure.

The molecular weights ^{and the} polydispersities ^were

measured by light scattering ^{and GPC on the deu-}

terated Polystyrene precursor and ^on the final pro- duct. The characteristics of the samples ^{are listed in}

table I.

Six copolymer ^{solutions were} prepared ^{in the}

range 2.46 x 10-2 _

c (g.cm- 3 ) _

^{50.8 x 10- 2. The}

solvent used was a mixture of deuterated and

hydrogenated ^benzene

(51.3/48.7)

^{which corres-} ponds ^{to the zero} average ^contrast conditions. The

overlap concentration for this system is of the order of 0.14

g.cm-3.

^SANS measurements on the diblock

copolymer ^{solutions were recorded at room} tempera-

ture on the D17 spectrometer ^{at ILL}

(Grenoble- France).

^{Details on the} experimental ^set up ^are

given ^elsewhere

[8].

^The sample-detector ^distance

was 1.4 meter with a neutron wavelength ^{of 12} A

allowing

^a range of scattering vectors q ^between 1.6 x 10-2 and 14 x

10-2 (A-’)

^to be covered. All the scattered intensities were corrected for trans- mission and incoherent background according ^to

classical data treatments.

Table I. ^-

Polymer

characterization.

Measured by ^a- GPC b- SANS c- NMR ; d- Calculated from GPC and taking ^account of the axial dispersion.

3.2 RESULTS AND DISCUSSION. ^- The SANS results ,are presented ⁱⁿ figure 4 where the variation of the

Fig. ^{4. -} Normalized static neutron scattering ^functions

of the diblock copolymer PSD-PSH 561 in benzene

H/D mixture in the zero average ^contrast limit.

Copolymer concentrations

C (g. cm-3):

^{+ 2.46 x}

10- 2 ;

x 4.95 x

10- 2 ;

^c3 0.101; 00.150 ; . 0. 302 ; ^A 0.508. Full line : theoretical curve calculated following equations (16)-(18).

intensities scattered by ^the solutions, normalized by

the concentrations, ^are

plotted

^{as a function of the}

wavevector. The results evidence that the observed

scattering ^{curves are} superposed

taking

^{account the} experimental accuracy. Such ^a type ^{of behaviour is} predicted by ^the theory

(see

Eq.

(15)

^and

Fig. 3).

^In

our experiments ^at low concentration

(c

^{= 2.46 x}

10-2g.cm-3) vcpN=2A2McQ::0.6

^{while at}

high

concentration

(c

^{= 0.508}

g.cm- 3)

^{vpN ==== 12.2}

where

A2 is

^the second virial coefficient as measured

by light

scattering (A2

^{= 1.2 x 10-3} cm3 .

g-l. mole-1).

^{The other remark which can be}

made about figure ^{4 is that} the scattered intensity

exhibits a maximum, ^the position of which is inde-

pendent ^{of the} concentration. In the same

figure

^we

have also drawn the theoretical variation of the normalized scattered intensity ^{as a function} of q.

This variation has been calculated using

equation (15)

by assuming ^that both sequences obey ^Gaussian

statistics. In order to take

polydispersity

^{into ac-}

count, it is also assumed that both sequences ^are

independently polydisperse

and have the Zimm- Schultz distribution

[9].

With these conditions

equation (15)

^can be written:

where

N n

is related to the number average molecular

weight ^{and :}

with

Rn and a n ^are respectively the number average radius of gyration ^and composition

[10].

^{In the}

numerical calculations we have used the value of the radius of gyration ^{of the} copolymer given by ^the

relation :

which is valid for PS in the theta conditions as well as

for good solvents in the low molecular weight range

[11].

The number average radii of gyration ^and

compositions

have been estimated in the frame of a

Gaussian statistics and

using

the values

appearing

ⁱⁿ

table I. As seen from

figure

^{4 the} agreement ^between the

experimental

results and the calculated scattering

function is

quite

satisfactory. ^{In fact the influence of}

the

polydispersity

^{on the scattered} intensity ^{is well}

known and results in a shift of the maximum towards

lower q values while the peak is broadened with respect ^{to a} monodisperse sample.

From a theoretical point ^{of view the}

assumption

^of

Gaussian statistics

adopted

^{in the} calculation seems to be very crude. This remark holds especially ^for

the lowest concentration because it is well known that in dilute solution the chain conformation is

perturbed by excluded volume effects. Nevertheless the experiments ^{have been carried out on a small}

molecular weight polymer for which the mean di- mensions are

quite

^{close to the} unperturbed ^dimen-

sions. Furthermore, the range of scattering ^vectors investigated corresponds ^to

qRG

values smaller than 4 and the results reported by ^Oono

[12]

^show that,

below this value, the static scattering factor of the chains in the good solvent limit does not differ

significantly

^{from the} Debye ^function.

1968

4. Conclusion.

In this paper ^{we have} discussed the

scattering

behaviour of diblock copolymer ^{solutions in the}

frame of the Random Phase

Approximation.

^We

have focused our attention on two limits namely ^the

limit where the solvent matches one of the sequences and in the zero average ^contrast limit. In these cases

the theoretical developments ^{lead to}

simple

express- ions which are easy ^to discuss. Interesting ^situations

for further

experiments

^{are found. The} preliminary

small angle ^neutron

scattering

^results

presented

^here

show that the proposed theory ^{can be used in a wide}

range of copolymer concentrations. The present study confirms that the RPA is also well adapted ^to

the case of copolymer ^solutions.

Acknowledgements.

We want to thank Dr. A. Rennie for his assistance

i

during ^the measurements on the SANS instrument D17, ILL, Grenoble, ^France.

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