Comb-like macromolecule conformation in smectic phase. Interpretation of neutron-scattering data

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Comb-like macromolecule conformation in smectic phase. Interpretation of neutron-scattering data

A.B. Kunchenko, D.A. Svetogorsky

To cite this version:

A.B. Kunchenko, D.A. Svetogorsky. Comb-like macromolecule conformation in smectic phase.

Interpretation of neutron-scattering data. Journal de Physique, 1986, 47 (12), pp.2015-2019.

�10.1051/jphys:0198600470120201500�. �jpa-00210396�

2015

COMB-LIKE MACROMOLECULE CONFORMATION IN SMECTIC PHASE.

INTERPRETATION OF NEUTRON-SCATTERING DATA A.B. KUNCHENKO* and D.A. SVETOGORSKY

*Laboratory of Neutron Physics

**Laboratory of Theoretical Physics

Joint Institute for Nuclear Research Dubna, ^{Head Post} Office, P.O. Box 79, 10 10 00 Moscow, U.S.S.R.

(Regu ^{te 20} juin 1986, ^{r6visg Ze ler} octobre, accepté ^{Ze 7} octobre 1986)

Résumé : Des considérations théoriques ^sim- ples permettent ^{de définir la} conformation de la chaîne principale d’une macromolécule

en peigne ^{sur la base des} spectres ^{de dif-}

fusion de neutrons aux petits angles. ^La chaîne macromoléculaire en phase smectique

est représentée par des pelotes quasi-bidi- mensionnelles placées ^au hasard dans les différentes couches smectiques. ^{La taille}

de ces sous-chaînes ne dépend pas de ^la

masse moléculaire globale ^du polymère. ^Les

sous-chaînes bidimensionnelles sont reliées entre elles par des passages à ^{travers les}

couches. Elles peuvent ^être considérées

comme des défauts de la phase smectique. ^Le modèle proposé ^ici permet ^de calculer le nombre de ces défauts à partir ^du rayon de giration ^{de la} macromolécule smectique ^tel qu’il ^est mesuré par la technique ^{de diffu-} sion de neutrons aux petits angles.

Abstract : Simple theoretical considera- tions allow us to define the conformation of the macromolecule main chain from neu-

tron experiment ^{data. The} macromolecular chain in the smectic phase is divided into

quasi-two-dimensional ^coils randomly placed

in different smectic layers. ^{The size of} these coils is independent ^{of the} polymer

molecular weight. ^The two-dimensional sub- coils are connected with each other by ^a

number of crossings ^{of the} layers. They ^can

be considered as defects of the smectic

phase. ^{The model} proposed ^{here allows us to} calculate the number of defects from the radius of gyration ^{of the smectic}

macromolecule which was measured by ^small angle ^neutron scattering.

LE JOURNAL DE PHYSIQUE

J. Physique ⁴⁷ (1986) ^{2015-2019 DÉCEMBRE 1986,}

Classification

Physics ^Abstracts

61.30 - 61.40K

’

There has been growing ^{interest in the} study ^{of the} liquid-crystalline (LC) ^state of polymers in different mesophases [1-3].

The method of selected deuteration provides direct information on the conformation of the macromolecule in the bulk from neutron

scattering ^data.

First measurements of the main chain ra-

dius of gyration in the nematic and smectic

mesophases performed by ^small angle ^neutron scattering (SANS) ^are presented ⁱⁿ [4,5].

In [6] ^the interpretation of the neutron

scattering ^{data for the LC} polymer ^{in the} nematic phase ^is given.

This paper deals with the interpretation

of presently ^available data of neutron

scattering ^{on smectic} polymers ^from [5].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470120201500

2016

1. Experimental ^data 5 .

In the paper [5] the chain of the type :

has been investigated by ^SANS and neutron diffractions (ND). ^{The molecular} weight ^is

3 ^x 10‘5 Dalton. The macromolecule of such

a polymer ^{contains 680 monomers and has a} contour length ^{1700 A.}

The measured values of the projections

of the radius of gyration ^{for the macromo-}

lecule in the smectic phase ^{are :}

where R. ^and Ry ^{are the} projections ^{of the}

radius of gyration ^on the directions per- pendicular (x) ^and parallel (y) ^{to the}

smectic layer plane.

The total radius of gyration ^{of the ma-}

cromolecule in the smectic phase ^{can be} expressed by projections :

The measured value of the radius of gyra- tion in the isotropic phase ^is

The comparison ^between (2) ^and (3) ^shows that the rigidity ^{of the} macromolecule is

greater ⁱⁿ the smectic than in the isotro- pic phase.

The thickness of the smectic layer ^{D is} measured by ^{the ND method}

2. The Model of an anisotropic freely jointed ^chain.

The large anisotropy ^{of the} projections

of the radius of gyration ^{of the LC} polymer macromolecular chain in the smectic phase

is the result of the difficulties in cross-

ing through ^the mesogenic ^{layer. It is}

natural since the smectic phase ^{can be con-}

sidered as a microphase separation ^of mesogenic ^and aliphatic parts [7].

Let us refer to the part ^{of the chain} intersecting ^{the smectic} layer ^{as the} crossing segment. ^{It is} reasonable to

assume that the crossing segment ^will straighten ^{and so will be} practically per-

pendicular ^{to the smectic} layer. ^The crossing segment ^{forms a defect in the}

smectic layer ^on the-molecular level. The detailed structure of this defect is unknown, though ^{the SANS} method, ^{as shown} below, ^{allows us} to evaluate the number of

such defects.

Let us construct the model of a freely jointed chain with the following proper- ties :

(1) The chain accomplishes random walks in the smectic layer plane ^{so that the} backbone of the macromolecule is located in the aliphatic layer. ^{Let us} describe the

flexibility ^{of such a} two-dimensional chain

by ^{the Kuhn} segment b,.

(2) ^{The chain} accomplishes random walks between the intermediate layers ^{with the} step equal ^{to the} inter/layer ^{thickness D.}

(3) ^Since crossing through the smectic layer is quite difficult, it must be taken into account by introducing different pro- babilities for the steps ^{of the} freely jointed chain in the plane and between the layers.

The simplest way to do it is to ^use the lattice model.

In this case the space distribution func- tion for one segment ^{of the} freely jointed

chain is :

where 6 is the Dirac delta function and p

is the normalized probability of the inter- layer crossing.

If p ⁼ 1/3, ^{then the} step probability ^is equal ^{for all} three directions.

As is well-known, ^{the mean} square distan-

ce between the ends of a freely jointed

chain is

where N is the number of chain segments, ^r

is the radius vector of one Kuhn segment

and (r)2 ^is given by :

Taking (5) ^and (7) into-account we obtain for the anisotropic ^{chain :}

For the mean square projections ^{of the}

distance between the ends of the chain we

obtain :

It is well-known that any ^real chain (without effects of excluded volume) ^{can be} approximated by ^an equivalent freely join-

ted chain which has the same contour length

and average distance between the ends as

those of the real chain.

The magnitude ^{of the Kuhn} segment ^{A and}

the number of such segments in the chain

are defined as :

where L is the contour length of the chain.

In the case when the chain is character- ized by ^two segment bl ^{and D, we find N}

from :

And using (5) ^and (7) ^{we obtain :}

rhe radius of gyration depends ^{on the dis-}

tance between the ends of the chain ^{as :}

Therefore, ^{for the} projections ^{of the}

radius of gyration ^{we have :}

In equations (12) , ^{(14) and} (15) ^the quantities Rx, Ry ^{and D are} determined ex-

perimentally, ^{the contour} length ^{L is a}

known quantity, and the chain parameters N,

b, ^{and p are unknown} quantities.

It must be taken into account that the real chain performs ^{random walks} not stric-

tly ^{in the} plane, but within the whole intermediate layer ^{of a} finite thickness.

These random walks will not affect the R,

value owing ^{to the small} thickness of the intermediate layer, ^but they ^{must be}

accounted for in the expression ^{for the}

contour length (12).

Let us denote the contour length ^{of the}

chain outside the plane by A L,. ^{It is rea-}

sonable to suppose that :

where A is the mean contour length per Kuhn

segment ^{of the} freely jointed in the inter- mediate layer.

With (16) taken into account equation (12) ^{becomes :}

where A is an unknown quantity. ^{Let us}

suppose that it is of the order of the thickness of the intermediate layer (5-10 A).

For convenience we introduce in equations (14), (15) ^and (17) ^the following ^nota-

tion :

then these equations ^{become :}

N, ^and N1 ^{are the} average numbers of the chain segments ^oriented normally and pa- rallel to the smectic layer, respectively.

For the isotropic distribution :

The quantities R, ^{and D are measured ex-} perimentally, ^therefore N, ^{can be defined}

from (19) ^{as :}

Ni : this is the number of crossing seg- ments (defects) ^{for one} macromolecular chain. The quantity

is a characteristic of the smectic phase.

Now let us estimate the accuracy of the

quantity ^k. Expression (24) ^{is obtained un-}

der the assumption ^{that the} crossing

segment ^is stretched ^as far as possible,

2018

i.e. it has transconformation. Another

inaccuracy ⁱⁿ (24) reflects the indefinite thickness of intermediate and mesogenic layers. Nevertheless, ^all this does not in- fluence the quantity N//.

Using equations (20) ^and (21) ^{we define}

where :

The quantities bl ^and N1 ^are defined with

an accuracy of the second order of the pa- rameter A/bl. ^{At last, let us} express p in terms of new variables :

3. Calculation of the parameters ^{for the} model chain.

Using experimental data from [5] ^and (23)

we obtain :

This is the number of crossing segments

for one macromolecular chain of the polymer with a given ^molecular weight. ^From (24) ^we

obtain for k : t

i.e. crossing segments amount to approxima- tely ^{6 %} of the total chain length.

N, crossing segments divide the chain into N, ^{+ 1} quasi-two-dimensional ^subcoils.

A typical conformation of the macromolecu- lar chain of the smectic LC polymer ^is shown in figure ^1.

The mean square distance between the ends of such a subcoil Chy)z ^is independent ^of

its internal structure and we can write down :

where Ry ^{is the} experimentally ^measured projection ^{of the radius of} gyration ^for

the whole coil. Using (31) ^and from (1) ^we obtain :

Correspondingly ^{for the} projection ^{of the}

radius of gyration and the chain length ^of

one subcoil we obtain :

- .

where Li is the average length ^{of the sub-}

coil chain. (In general, h1y ^and Ll ^largely

vary since crossing segments appear ^occa-

sionally ^{for the real} subcoil). ^{Note that} the quantities ⁱⁿ (32) ^and (33) ^{are inde-} pendent ^{of the} uncertainty ⁱⁿ A, ^while bl,

N and p ^as is seen from (25), (26) ^and (28), depend ^{on A. Here are these} quanti- ties for two values of A :

As is seen from (34) ^and (35), values of

bi, ^{N and p depend rather weakly on the pa-}

rameter A.

Let us take bl - ^{60 A . The comparison}

between bl ^{and the Kuhn segment for the}

coil in the isotropic phase (39 A) ^indica-

tes a considerable increase of the chain

rigidity ⁱⁿ the two-dimensional subcoil. It is caused primarily by ^the decreasing ^num-

ber of gauche-somers ^{in the} quasi-two-di-

mensional chain and by effects of excluded volume appearing ^{in this case [8].}

In the three-dimensional case, according

to the Flory theorem, the chain behaves as

in ideal one and effects of excluded volume can be neglected.

It follows from all this that it is desi- rable to measure the Kuhn segment directly by ^the Kratky technique [9]. Measuring only

Ry ^is not sufficient for a precise ^defini- tion of b-L ^{because of the} uncertainty ^{in A}

and the influence off excluded volume effects.

Fig. ^1.- Typical conformation of the macro- molecules backbone. Quasi-two-dimensional subcoil 3 placed ^{in the} intermediate layer 1, ^{are connected with} crossing segments ^4.

These segments create defects in the meso-

genic layer ^2. (Experimental ^{data are taken}

from [5]).

4. Temperature effects [10].

Studies of the temperature dependence ^of

the projections ^{of the} macromolecule main chain radius of gyration ^can give ^informa-

tion on the nature of defects of smectic comb-like LC polymers.

Supose that the smectic phase ^{is in the} equilibrium state, i.e. there is a dynami-

cal balance in,the processes of appearance and disappearance ^of smectic defects at a

given temperature. ^{In this case the} proba- bility ^{of defect} appearance p ^can be written down as follows :

where R is the molar gas constant, ^{T is the} temperature ^{and e- 6FIRT} is the statistical

weight ^of appearance of the crossing segment.

Estimating ^with (36) ^the energy ^{of a} smectic defect E (© F N E) at T = ^{330 K} for p ⁼ 0.01 obtained in (34) ^{we have :}

The temperature dependence ^of p (with known E) ^{is :}

and using ^{(14) we} obtain the temperature dependence :

The polymer considered is in the smectic phase ^{in the} temperature range from 45°C to 110°C. For this temperature difference, ^we obtain, using (39) :

---- , -m in ,

Thus the equilibrium theory predicts ^{a 10 %}

increase of R. ⁱⁿ the smectic temperature

range.

Note that the defect energy ^E is composed

of the segregation energy of mesogens and aliphatic parts ^of the macromolecule and the energy ^of the main-chain bending.

5. Conclusions.

According ^{to the} assumptions of the pre- sent paper random walks of ^a polymer ^chain

can be divided into two types - ^random walks in the smectic layer ^{and those} between layers ^In the latter case the step of random walks equals the thickness of the smectic layer. Therefore, measuring ^the

projection of the radius of gyration ^{of the}

macromolecule in the direction x (Fig. 1) and using (23) ^{one can determine} the number of crossing segments (defects) N,,.

The crossing segments divide the macromo- lecular chain into quasi-two-dimensional parts with the conformation of a statisti- cal coil. The average ^{size of these} subcoils can be defined using (31) ^{and mea-} suring ^the projection of the radius of

gyration ⁱⁿ the y -direction.

Absence of data on the influence of excluded volume effects and random walks of the main chain in the x. -direction in the range of the intermediate sublayer prevent

a precise definition of the Kuhn segment bl

in the plane. Therefore, ^it is desirable that the Kuhn segment (or ^the persistence length) ^{should be} directly measured in the smectic layer plane by ^the Kratky technique.

We hope ^{that the use of} small-angle ^neu-

tron scattering and neutron diffraction along ^with other methods (X-ray ^diffrac- tion, ^NMR etc.) ^{will soon} give ^a complete

idea of the comb-like LC polymer ^structure.

Acknowledgments

We are grateful ^{to V.P. Shibaev, L.B.}

Stroganov, ^R.V. Tal’roze and S.G. Kostromin from the Moscow University ^{for numerous}

discussions.We also thank our colleagues

Ju. M. Ostanevich, ^V.K. Fedyanin, ^J.

Plestil and V.B.Prieszhev for discussions

and critical comments.

References

[1] Liquid Cristalline Order in Polymers,

edited by ^{A. Blumstein} (Academic press, New-York, London) ^1978.

[2] ^Shibaev V.P., ^Plate N.A., ^Adv. Polym.

Sci. 60/61 (1984) ^173.

[3] Polymer Liquid Crystals, ^edited by

Ciferri A., Krigbaum ^W.R., Meyer ^R.B. (Aca- demic press, New-York, London) ^1982.

[4] ^Kirste R.G., ^Ohm H.G., ^{Makromol. Chem.}

Rapid ^{Commun. 6} (1985) ^179.

[5] ^Keller P., ^Carvalho B., ^Cotton J.P., Lambert M., Moussa F. and Pépy G., ^J. Phy- sique ^{Lett. 46} (1985) ^L-1065.

[6] ^Kunchenko A.B., Svetogorsky D.A., ^Sub- mitted to Liquid ^Cristals.

[7] ^{de Gennes} P.G., ^The Physics ^of liquid

Crystals (Oxford ^Univ. Press, ^{London and} New-York) ^1974.

[8] ^{de Gennes} P.G., Scaling Concepts ⁱⁿ Polymer Physics (Cornell ^Univ. Press, Ithaca and London) ^1979.

[9] Kratsky O., ^Pure Appl. ^{Chem. 12} (1966)

483.

[10] ^The probability p of interlayer ^tran- sition with a temperature dependence ^simi-

lar to (38) ^{was also used} by ^{W. Renz and M.}

Warner in Phys. ^{Rev. Lett. 56} (1986) ^1268.