Evidence by X-ray scattering of defects in the lamellar stacking of the SmA phase of a side-chain polymer

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Evidence by X-ray scattering of defects in the lamellar stacking of the SmA phase of a side-chain polymer

P. Davidson, A.M. Levelut

To cite this version:

P. Davidson, A.M. Levelut. Evidence by X-ray scattering of defects in the lamellar stacking of the SmA phase of a side-chain polymer. Journal de Physique, 1988, 49 (4), pp.689-695.

�10.1051/jphys:01988004904068900�. �jpa-00210744�

Evidence by X-ray scattering of defects in the lamellar stacking

of the SmA phase ^{of a} side-chain polymer

P. Davidson and A. M. Levelut

Laboratoire de Physique ^{des Solides, Associé au C.N.R.S., Bât. 510,} Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay,

France

(Reçu le 28 octobre 1987, accepté ^{le 4} janvier 1988)

Résumé.

Nous avons effectué une expérience ^de diffusion des _{rayons X} sur un polymère mésomorphe ^en peigne (polyméthacrylate) ^en phase smectique A. L’allure générale ^{du cliché est} classique ; toutefois, ^on repère l’existence d’intensité diffusée localisée sur des lignes inhabituelles. Nous expliquons l’apparition ^de

cette intensité par la présence ^de défauts dans la phase SmA. Ces défauts pourraient ^{être les zones où les}

chaînes principales ^sautent d’une couche à la suivante.

Abstract.

An X-ray diffraction pattern ^{in the} SmA phase ^{of a} mesomorphic ^{side chain} polymethacrylate displays ^some unusual diffuse lines in addition to the elements already described. We account for these lines by introducing ^{some defects which disturb the lamellar order.} Such defects may ^{be the} places ^{where the} polymer

main chain hops ^{from one} layer ^{to an} adjacent ^one.

Classification

Physics ^Abstracts

61.30J

61.40K

Introduction.

A mesomorphic side chain polymer of formula :

has recently ^been synthesized [1].

This polymer presents ^the following polymor- phism : ^{G 50} S A ^{95 N 110 I where G} stands for glassy S A ^state (Tg = 50°C).

Its organization ^{in the} SmA phase ^has been studied

by X-ray diffraction and it was possible ^{to infer that}

the main chain is strongly anisotropic and confined between the layers ^{of the} mesogenic ^{cores. In order}

to check this result, ^{the same} polymer deuteriated

on the backbone was synthesized and studied by

neutron diffraction [2].

This deuteriated polymer (called PMD) gives ^the

same X-ray diffraction patterns ^{as its} hydrogenated homologue (called PMH) ^{but with a} slightly ^better

contrast. This has enabled ^us to notice some unusual diffuse lines hardly detectable ^on the X-ray patterns of PMH.

In this paper, ^we shall discuss the structural features which _may give ^{rise to} this diffuse scattering.

Experimental procedure.

The X-ray diffraction experiments ^are performed

with the same apparatus ^as described in refe-

rence [1]. ^The sample is held in a Lindemann

capillary ^{of 1.5 mm diameter} placed ^{in an oven itself}

heated by ^{an air stream. The} temperature ^is kept

constant within 1 °C. An aligned sample is obtained

by applying ^a magnetic field of 1.7 T for a few hours at a temperature just ^{below the} nematic-isotropic

transition. A monochromatic point focussing X-ray

beam of wavelength ^{A = 1.5405 A} is obtained by

reflection on a doubly ^bent pyrolytic graphite ^mono-

chromator. The diffracted X-rays ^{are collected on a} cylindrical ^{film at 60 mm from the} sample. ^{In this} geometry, the direct and reciprocal spaces present ^a cylindrical symmetry around the magnetic field direc- tion.

Figure ^{1 shows an oriented} X-ray diffraction pat-

tern of PMD taken at room temperature ^{in the} glassy SmA ^{state. We find} once more the same

elements ^as those observed for PMH and discussed at length ^{in reference} [1]. ^We shall here briefly

review these elements and remind the reader of their

interpretation :

- The Bragg spots (a) along the meridian are

resolution limited ; they ^{indicate a} lamellar order of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004904068900

690

Fig. la.

X-ray diffraction pattern ^{of PMD in the} glassy SmA ^state.

Fig. lb.

^Schematic representation ^{of the} diffraction pattern : (a) Bragg spots ; (b) Large angle ^{diffuse cres-}

cents ; (c) ^{Diffuse lines ;} (d) ^Diffuse spots ; (e) ^Diffuse streaks ; (f) Moustaches. (The ^{inner circle} going through

the lst Bragg spots ^is only ^{due to the small} disorientation of the sample.)

periodicity d

^{29.5 A which is} roughly equal ^{to the}

size of a monomer f = 28 A measured on Dreiding

stereomodels.

-

The wide angle ^crescents (b) perpendicular ^to

the equator show that the mesogenic ^{cores are}

oriented along ^the magnetic ^field direction Oz and that the layers ^are liquid-like. ^{Therefore we have a} SmA phase.

-

A set of equally spaced diffuse lines (c) perpen- dicular to the meridian corresponds ^to the intersec- tion with the Ewald sphere ^{of a set of} parallel ^and equally spaced reciprocal planes. This scattered

intensity ^{comes from} uncorrelated columns of meso-

genic ^{cores which are out of their mean} positions along ^Oz.

-

Some diffuse spots (d) show the existence of a transverse displacive distorsion (some ^{sort of undula-}

tions of the layers).

-

A diffuse streake for PMD or spot ^{for PMH} indicates the presence of ^a weak antiferroelectric

ordering which doubles the period along ^Oz.

In figure 1, the elements (a) ^and (b) ^are greatly overexposed ^{so that one can} clearly ^{see an additional}

diffuse streak (f) going through ^{the lst} Bragg spot and inclined at an angle 0 ^{= 15° with} respect ^{to the} equator ^Ox. Moreover, ^{this diffuse} streak is not

centrosymmetrical around the Bragg spot. Actually,

oriented patterns obtained with PMH also present these diffuse streaks when closely inspected, ^but

their appearance is blurred compared with those

presented by ^{PMD. This is} probably ^{due to the fact}

that the deuteriated and hydrogenated chains have different elastic properties. Indeed, ^the physical properties ^{of the two} compounds ^{such as} the transi- tion temperatures ^are slightly ^different [2].

Optical diffraction on models of defects.

We shall now try to account for the scattered

intensity (f) by introducing ^some defects in the

SmA phase.

First, ^{let us} consider in the lamellar structure, ^an

object ^{with the} shape ^{of an} ellipsoid ^of principal

axes Ox’, Oy’, Oz’ rotated by ^an angle ^{0 with} respect

to the axes of the lamellar structure (cf. Fig. 2a),

with principal ^{diameters a, b, c. Let us assume that}

this object presents ^a periodic density ^modulation

with period ^d’ = d _along Oz’. Then the diffusion cos 0

pattern ^{of such an} object ^{will be a series of} ellipsoidal

domains centred on points equidistant by 1, _d’ ^along

direction Oz’ itself rotated by ^{the same} angle 6 ^with respect ^{to Oz in the} reciprocal space [3] (cf. Fig. 2b).

These domains will have for principal ^diameters

11 .1

1, 1 ^{and 1.}

a b b c

Now, ^{if we have a random} assembly ^{of such} N objects in the structure, then they ^{do not diffract}

Fig. ^2.

a) Representation ⁱⁿ real space of the object ^as

an ellipsoid ^{of radii a,} b, ^c and tilted by ^an angle 0 ^{in the} x’ 0 ^z plane. b) Corresponding diffusion domains in recipro-

cal space.

coherently (i.e. ^{there are no} interferences between

them) ^{and the scattered} intensity is N times the scattered intensity ^{of one} of them. These objects

may be called defects since they ^occur randomly ⁱⁿ

the structure.

At this point, ^{we must notice that} though ^{there are}

defects in the structure, ^the Bragg reflexions remain resolution limited. This means that the defects do not break the smectic planes ^{which remain conti-}

nuous.

Since the defect is inclined by ^an angle 0 ^{with the}

lamellar structure, ^{its cross} section in the plane Ox’ y

cannot be a disk (a =#= b ), in other words the object

must be biaxial. The problem ^{now is to know}

whether a is close to b, ^{and then we have a line-}

defect or much bigger ^than b, ^{and then we have a}

wall-defect. At this point, ^{we must} remember that the sample presents ^a cylindrical symmetry around Oz. Then, ^{our defects are} distributed on a cone of axis Oz and angle ^{0. This} symmetry ^forbids

us to measure a and b independently ^{and we cannot}

decide whether we have line defects ^or wall defects.

In order to probe ^{the nature of these defects, we}

have performed ^some optical diffraction experiments

on a few models of defects. For this purpose, ^we have used ^a small He-Ne laser of wavelength

À

633 ^nm and ^a transverse mode selector [4].

Figures ³ (a, b, c, d, e, f) ^{show some} sketches and their diffraction patterns (with ^{such a} method, ^we only have 2-dimensional representations ^{of the struc-}

ture, ^we have no information on the third direction.

Fig. ^3.

Various sketches of defects and their optical

diffusion patterns : (a) ^nematic (no ^internal structure) ; (b) ^{a = 0} (scattered intensity ^maximum along Oz’) ; (c)

a = - 0 (scattered intensity maximum away from Oz).

This sketch is in agreement ^{with the} X-ray ^diffusion

pattern ; (d) a ^{is an} increasing function of 8 ; (e) xo

(apparent width of the defect) ^{is a} decreasing function of

02013 small curvature of the moustache ; (f) ^d’ (internal

period ^{of the} defect) ^{is a} decreasing function of 0. We

observe, here too, ^a small curvature of the moustache.

692

This meets the experimental fact that ^we cannot

distinguish ^{Ox’ and} Oy because of the uniaxial symmetry ^around Oz).

Firstly, ^{we have} examined the diffraction pattern of an object ^{with no internal} structure - i.e. : a

nematic zone (Fig. 3a). We observe some diffuse lines going through ^the Bragg spots and inclined by

(J = 15°, ^but they ^are symmetrical ^with respect ^{to the} Bragg spots. (In figure ^{3a, we have} represented

several defects in order to obtain a stronger intensity

and the diffuse line is slightly ^modulated along ^Ox’

by ^some lateral interferences among defects.) ^We

can justify ^{this in the} following way : the presence of holes in the SmA matrix induces the appearance of

some diffuse scattering around the origin ^{of the} reciprocal space but since the SmA ^matrix presents ^a periodicity, this diffuse scattering is observed around each Bragg reflexion. Since the defects are broad

along Oz’ and thin along Ox’, the reflection domains

are thin along Oz’ and broad along ^Ox’.

We now want the diffuse line not to be symmetri-

cal with respect ^{to the} Bragg spot and this leads us to introduce a diffracting pattern ⁱⁿ the defect. Such a

pattern ^must repeat ^itself periodically in the defect with a period ^d’ along Oz’ such as d’ = d since

cos (J the diffuse lines (f) ^must go through ^the Bragg spots

(a). ^The simplest pattern possible ^{is a} straight ^line segment which makes ^an angle ^a with Ox and which may present ^a phase ^shift cp with the layers (Fig. 4).

(Thus, ^a

^{0 and cp}

⁰ correspond ^{to the} disap-

pearance of the defect).

Fig. ^4.

^{Internal structure of the defect} used for the diffraction calculation. Dashed line : missing ^smectic layer ; ^{Solid line : tilted} segment playing the role of antenna. a: cp = 7T ; b : cp = o.

Qualitatively, ^these segments behave like anten- nas, they ^diffuse mainly ^{in a direction} perpendicular

to theirs. In this way, ^{we can} maximize the scattered

intensity ^{in a direction} perpendicular ^{to a. In} figure ^{3b, for} instance, ^the intensity is maximum

along ^{Oz’ and we} find that the diffuse line is not

symmetrical anymore with respect ^{to the} Bragg

spot ; ^it points towards the small angles rather than the wide angles. ^In figure 3c, ^the diffracting segments

are set at an angle such that the scattered intensity ^is

maximum _away from the direction Oz’ of the defect.

In that case, the diffuse line points towards the wide

angles rather than the small angles. ^{In order to}

obtain a second set of diffuse lines symmetrical ^to

the first one with respect ^{to the} meridian, ^we would

just ^{have to} represent ^{a second set of} similar defects inclined by ^an angle - ^0.

We find that, ^{in this} case, the optical diffraction pattern corresponds ^{well to} the actual X-ray ^diffrac-

tion pattern.

A simple analytical ^treatment of this model is

given ^{in the} appendix ^and gives ^an estimated value for the apparent width of the defect : xo

^{20 ± 5 A.}

This apparent ^width represents ^a complicated ^func-

tion of a and b, values that we cannot measure

independently. The width of the diffuse line (f) along Oz’ is close to the resolution of ^our X-ray ^set

up. This ^means that the length ^c of the defects is

roughly ^{estimated as a} few hundreds of Angstroms.

If we now come back to the X-ray diffraction pattern, ^we may notice that the diffuse line (f) ^is slightly curved towards the meridian Oz. This may remind the reader of a similar effect predicted by ^De

Gennes [5] ^{for the} light scattering by smectics : the maximum intensity ^scattered by ^the elastic defor- mation fields of ^some edge dislocations should be located ^on parabola-shaped diffuse lines of equation sz K BK ^{B S2 . -L The elastic constants Ki of some side}

chain polymers ^have already been measured [6] ^and

found to be comparable ^to those of conventional nematics (Ki ^{= 10-11 in S.I.} units). ^{If we now}

assume that B were equal ^{to that of a} conventional smectic B = 107 ^{in S.I.} units), ^{we would attain a}

ratio K _ 10 _B ^{A in the} range of the X-ray spectra.

However, this linear elasticity ^{model is} essentially

valid far from the core of the defect, ^{i.e. at small}

diffusion vectors, very close ^to the Bragg spot ^and therefore we need to give ^a description ^{on a smaller} scale, typically ^{a few 10 A.}

A simple way ^{to account} for this small curvature is to consider that the angle 0 ^of the defect may vary from place ^to place. ^{If the} angle 6 of the defect varies, ^its structural parameters ^{such as} xo a ^{and d’}

are very likely ^to vary with ^0. (Remember ^that xo ^{is the apparent} width of the defect, ^{« is the} angle

between the scattering segment ^{of the} pattern ^and the smectic layers and d’ is the period ^{of the} patterns along ^the defect) (cf. Fig. 4). ^{In order to check the}

effects of such variations, ^{let us} approximate ^the

curved diffuse line (f) by ^a polygonal ^line composed

of straight ^line segments. ^{This means that we con-}

sider a defect as composed of several sections which

make different 0 angles with the smectic matrix.

First, figure ^{3d shows a case where a is an} increasing function of 0 but the optical diffraction pattern ^{shows a} fan of diffuse lines rather than a

curved line.

Now, ^{let us consider} xo ^{as a} decreasing function of 0 1 is the width of the reflection domains. These

Xo

sections of the defect which have a small 0 will lead to a small horizontal diffuse line while those sections which have a bigger ⁰ will lead to a long ^inclined

diffuse line. This situation is sketched on figure ^3e

and the resulting optical pattern shows indeed a

small curvature effect.

If the period d’ ^{of the} patterns along ^{the defect is a} decreasing function of 0, then these sections which have a small 0 will present ^a larger period ^{and their}

reflection domains will be shifted along Oz’ towards the small angles. Conversely, those sections which have a larger 0 ^{will see} their reflection domains shifted towards the wide angles. This situation is sketched on figure ^{3f and we can} also notice a small curvature effect on the diffraction pattern.

Finally, ^we may say that the small ^curvature of the diffuse line (f) ^comes from the fact that the direction of the defect varies from place ^to place ^{and with it,}

the apparent ^width xo ^{and the} period d’ ^{of the} patterns ^inside. Anyway ^{this curvature is hard to}

notice ^on the X-ray diffraction pattern and therefore

we should not overemphasize ^this point.

Discussion.

So far, ^{we have} only ^{drawn from the} X-ray ^diffrac-

tion pattern ^{as much} information ^as possible ^about

the objects ^which give ^{rise to} the scattered intensity (f). ^{We have not} made any assumption ^as regards

the physical ^{nature of these} objects. ^{We shall now} present ^a hypothesis ^{upon the} origin and constitution of these defects.

This kind of diffuse scattering ^{has never been}

observed in the X-ray pattern ^{of small} mesogens which means that in some way, it must be linked to the presence of the polymer backbone. In a

SmA phase, the backbone is confined between two

layers ^of mesogenic ^{cores, in a width of at most 10 A} along the normal (Oz ) ^{to the} layers [1]. However, the radii of gyration of the main chain along Oz(RU) ^and perpendicular ^{to it} (R1.) ^{have been}

measured in the SmA phase by ^small angle ^neutron scattering experiments [2] : RII

^{22 ± 3} A, _{R 1.}

86 ± 9 A. The comparison between the width of 10 A and Ril

^{22 A} shows that a polymer ^main

chain cannot stay strictly confined between two

layers ^of mesogenic ^cores. The smectic period being

29.5 A, ^{we see} that, roughly, each main chain must

hop ^{once from a} layer ^{to an} adjacent ^{one and thus}

creates a defect in the smectic plane. ^Such layer hopping by chains has been theoretically predicted

by Renz and Warner [7]. If these defects correlated themselves along ^{a direction} Oz’, they ^{would form}

an object very similar to what is sketched in

figure ^{3c. We have made a} tentative and schematic

drawing of the defect in figure 5 ; ^{it is to be} compared ^with figure 3c, which contains the best information that we could obtain from the diffuse line (f). ^{The reason} why ^the layer hoppings ^should

correlate along ^some direction is that the place

where a main chain crosses a layer ^{is a} region ^of

distortion where another defect is more likely ^to

appear. In other words, these defects can minimize their elastic energy by coalescing. (M. Warner, pri-

vate communication.) ^A rough estimate of the ratio of scattered intensities in the diffuse line (f) ^{and in}

I

the Bragg spot (a) ^is If =z 10- 4_10- 6. (This ^{ratio is} Ia

difficult to evaluate because Ia ^{is much} bigger ^than If and because the diffuse scattering ^{lies on a} cone.)

Fig. ^5.

Hypothesis ^{for the nature} of the defect. The spacers have been omitted for the sake of clarity. ^The

dashed loop represents ^the limits of the defect. The

straight ^line segments represent ^the mesogenic ^{cores. The}

thin curved lines represent ^the polymeric backbones. Only

a few layers ^{have been} represented.

When hopping ^{from one} layer ^{to another} (on ^a

distance 30 A), ^{a main} chain disturbs approximately

12 of its mesogenic ^cores which will diffract cohe-

rently. ^Our resolution is estimated to 10-3 A-1 and

our Bragg spots ^are resolution limited. If ^{we assume}

an area per mesogenic ^{core of} 20 Å 2, ^{then we have}

106 _mesogenic cores diffracting coherently. ^{In such}

20

a domain of radius 1000 A ^we shall have, ^on

average, ( ( 86 103 2 =102 polymer ^{main chains and as}

694

many defects. However, these defects do not diffract

coherently ^{and our} ratio of intensities should

roughly g y ^be equal q ^to (12 )2 x 102 = 10- 5 in correct

106 2

(106/20)2 ²⁰

agreement ^{with the} experimental ^data.

Conclusion.

We have shown the presence in the smectic matrix of defects which give ^{rise to} the scattered intensity

localized on the diffuse lines (f). ^These defects make

an angle 0

15° with the normal to the layers ^and

are several hundreds of Angstroms long ^{but we}

cannot tell whether they ^are line defects ^or wall defects. The defect must have ^a pattern repeated periodically along ^its main axis with ^a period d’

related to that of the smectic matrix d’ = d ) . ^{cos 0}

This pattern may just ^{be a} portion of smectic layer

tilted by ^an angle ^{a - 60° with} respect ^to the smectic matrix. A slight ^curvature in the diffuse line (f) ^is likely ^{to come} from the variation of the defect width

xo ^{or internal} period d’ ^{with its} angle ^0.

Finally, ^we think that the polymer ^backbones

when hopping ^{from one} layer ^{to an} adjacent ^one

create distortions which correlate themselves and build up these defects. This assumption is in fair agreement ^{with the} experimental evaluation of the

X-ray scattered intensities but needs to be checked

by electronic microscopy. ^It might ^be theoretically possible ^to relate the parameters of the defect such

as the tilt angle ^or the width by considering ^the entropic properties ^{of the} backbone, the elastic

properties of the smectic field and the nature of the spacer.

We are grateful ^{to Dr.} G. Durand and Dr.

J. Prost for helpful discussions and P. Ballongue ^for

technical assistance with the laser optical diffraction apparatus. ^{It is also a} pleasure ^{to thank Dr. P. Keller}

for kindly providing ^us with the deuteriated polymer samples.

This work has been supported by ^{a DRET contract}

nr. 85052.

Appendix.

We first want to calculate the form factor of the pattern represented ^on figure ^4a. By ^{a rotation of an} angle 0, ^{let us} place ourselves in a system ^{of axes}

(0, x’, z’) ^{attached to the} defect. The pattern ^is made by erasing the horizontal part ^{of the} layer (in

dashed line) ^and replacing ^it by ^a straight ^line segment (in ^solid line). ^The equations ^{of the two} segments ^{in our} system ^{are :}

-

dashed line :

-

solid line :

These two segments represent ^the electronic density

of the pattern compared with the smectic matrix and the scattered amplitude A (S) is the Fourier trans- form of this electronic density :

The result of this calculation is :

If we now remember that the patterns ^must repeat themselves in the defect, along ^{Oz’ with a} period

d’ = d , ^we shall have some scattered intensity only ^for Sz’

ⁿ cos 0 ^{with n} integer.

cos 0 ^z d g

Then, ^one easily calculates :

Figure 6 shows the profile I (Sx’ ) along ^the straight

line Sz,

Cos 0 _d (this ^{is the} equation ^{of the} straight

line on which the diffuse scattering (f) ^is located).

The parameters ^d

^29.5 A and 0 = 15° have been measured on the diffraction pattern ^and parameter

xo ^must take the value xo

^{20 ± 5 A in order to fit}

the experimental intensity profile ^{of the diffuse line} (f). xo ^{is an} apparent width of the defect, ^{function of}

a and b.

Figure ^{6 also} shows the position of the axis

Sz and therefore this model correctly ^{accounts for}

the fact that the line is not centrosymmetrical ^around

the Bragg spot.

For a simpler representation, this model impli- citely ^{assumes a} phase ^{shift of} cp

^7r between the

diffracting pattern and the smectic layer, ^{but it is} possible ^{to think of a} picture ^{with no} phase shift,

cp

0, ^{such as drawn on} figure ^4b.

The detailed calculations show that we cannot discriminate between these two cases because they mainly ^differ by ^their intensity ratio of the 1st to the 2nd diffuse lines (Sz’

cos 8 ^{and Sz,} = 2 cos 8 ) ^and

Fig. ^6.

Analytical ^{curve I} (Sx’) along S.,, = a7 : ^{d scattered}

intensity in the moustache versus S.,,. The second vertical axis represents ^the intercept with the meridian Sz. ^{One can} clearly ^{see that} I (Sx’) ^{is not} symmetrical ^with respect ^to this axis. ( B = 15° ; Xó

²⁰ A ; d

^29.5 A).

we have only observed the 1st diffuse line experimen- tally.

References

[1] DAVIDSON, P., KELLER, ^{P. and} LEVELUT, ^A. M., ^J.

Phys. ^{France 46} (1985) ^939.

[2] KELLER, P., CARVALHO, B., COTTON, J. P., ^LAM- BERT, M., MOUSSA, ^{F. and} PEPY, G., ^J. Phys.

Lett. 46 (1985) ^{L 1065.}

[3] ^For X-ray diffraction theory, ^{see for instance :} GUINIER, A., X-ray diffraction in crystals, ^im- perfect crystals ^and amorphous ^bodies (W. H. Frieman and Co, ^San Francisco).

[4] Advances in optical and electron microscopy 7, p. 17,

Eds. V. E. Cosslett and R. Barer (Academic Press, 1978, London, New York and San Fran-

cisco).

[5] ^DE GENNES, ^P. G., ^{C.R. Acad.} Sci. Paris B 275 (1972)

939.

[6] FABRE, P., CASAGRANDE, C., VEYSSIÉ, ^{M. and} FINKELMANN, H., Phys. Rev. Lett. 53 (1984)

993.

[7] ^RENZ, W. , WARNER, M., Phys. Rev. Lett. 56 (1986)

1268.