High resolution electron microscopy of liquid crystalline polymers

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I.G. Voigt-Martin, H. Krug, D. van Dyck. High resolution electron microscopy of liquid crystalline polymers. Journal de Physique, 1990, 51 (20), pp.2347-2371. �10.1051/jphys:0199000510200234700�.

�jpa-00212533�

High resolution electron microscopy ^of liquid crystalline polymers

I. G. Voigt-Martin, ^H. Krug and D. Van Dyck (1)

Inst. Phys. Chemie der Universität Mainz, Jakob Welder Weg 11, ^D6500 Mainz, ^F.R.G.

(1) Universiteit Antwerpen, Groenenborgerlaan 171, Belgium (Received ^{19 March} 1990, revised 31 May, accepted ^{7 June} 1990)

Abstract.

Special techniques ^of high resolution electron microscopy applied ^to liquid crystalline polymers in the smectic and discotic phases revealed characteristic distortions from the perfect

lattice periodicity ^{as well as defects. In order to} verify the direct relationship between electron

microscope images and structure, computer simulations were performed. Agreement ^{was found}

between simulated image ^{and structure} proposed ^on the basis of evidence from electron diffraction imaging. ^A theory ^was developed ^to explain ^this unexpectedly good agreement. Image

restoration by frequency filtering in reciprocal space ^was found to give good results for the discotic material.

Classification

Physics ^Abstracts

61.30E - 61.16D - 07.80

1. Introduction.

The potential ^{uses of} liquid crystal polymers ^{have focused on} applications in the fields of

optics, mechanics and electronics. Optical applications include information storage ^and optical displays ^{as well as} the manufacture of optical modulators and light ^wave guides [1].

The good mechanical properties arise from the high ^tensile strength ^{which can} be achieved

[2]. Recent interest has focused on the possibility ^of promoting charge ^transfer along ^the

stacked columns formed by ^{disc-like mesogens} using ^suitable doping [3].

All these propertiés ^are associated with the anisotropic ^{chemical structure of the} liquid- crystal forming mesogenic ^units [4]. Because of this anisotropy, specific spatial arrangements of atoms or molecules are promoted within the material [5], ^which finally give ^{rise to the} physical anisotropy. ^{The aim of structure} analysis by using diffraction techniques ^{is to obtain}

information about this spatial arrangement. ^When long range positional and orientational correlations ^are present, diffraction experiments using X-rays, ^{neutrons or} electrons enable a

precise determination of the atomic positions ^to be made. Such structure determinations have reached a high degree ^of sophistication [6]. ^A prerequisite ^{for a} proper ^structure analysis ^is

the availability ^{of a} sufficient number of diffraction maxima. Unfortunately, diffraction

experiments performed ^on liquid crystals usually ^{show a} dramatic loss of higher ^order

diffraction maxima [7].

For this reason diffraction experiments ^can only provide information about long range

spacings ^{or about the} average direction of the repeating units. Therefore it was felt that high

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510200234700

liquid crystalline phase. polymeric only synthesised during ^the past ¹⁰ years, dynamic processes ^are substantially slower and the l.c.

phase ^can be retained in the glassy ^{solid. For this reason it was felt} that these polymers represented ^{an ideal} opportunity ^for obtaining high resolution images ^{of the} solidified liquid crystalline phase. However, ^{in order to} understand and interpret these novel results correctly,

a comparison ^{with the same molecules} (monomers) ^{in the} crystalline phase ^was considered to be essential.

Two special types ^of samples (with very specific mesogenic units) ^{which can be} obtained in the crystalline (Al, A2) ^and liquid crystalline (B 1, B2) phase ^were chosen for this

investigation :

(1) Polymalone ^{ester with a} rod-shaped (calamitic) mesogenic ^unit (Azobenzene) (2) Triphenylene derivative with a disc-like mesogenic ^unit.

All the samples ^were prepared ^{in the} organic chemistry department ^{at Mainz} University

and their synthesis has been described in detail elsewhere [8, 9, 10]. ^{The chemical structures}

and phase transitions from isotropic (i) ^{to smectic} (s) ^and glass (g) ^or crystalline (k) ^are

indicated in table I.

In order to obtain meaningful results, it is essential to orientate the samples ^{so that the}

smectic planes and discotic columns can be viewed in the correct direction. We have published

details of our preparation technique previously [3, 4, 11, 13, 14] ^{and the} relationship ^between

the orientation of the mesogenic groups and the diffraction patterns ^{is shown} schematically ⁱⁿ

table II. The aim of this paper goes considerably beyond ^the previous ^{ones :}

(1) ^To compare high resolution images ^from crystals ^and liquid crystals ^{in order to find} specific differences between the two and to demonstrate the expected and well-known

crystallographic ^features of the former. This will also provide evidence that the novel features observed in the liquid crystals ^{are not} artefacts caused by ^radiation damage ^or sample preparation.

(2) ^To provide ^further evidence that the observed features in the images ^{are real. This}

involves careful analysis of the electron diffraction patterns, ^{which must contain some} evidence for the observed superstructural ^details.

(3) ^To improve image quality. The low dose required ^to image beam sensitive materials leads to an unfavourable signal/noise ^{ratio. Fourier} frequency filtering will be shown to

provide acceptable results for discotics, ^{if the correct} phases ^{are accounted} for, ^{but not for}

smectics.

(4) ^To provide ^a theoretical basis for the interpretation ^{of the} image. ^It will be necessary ^to show how the image ^{are related to the} projected potential ^{of the} object.

(5) To compare the proposed ^structure with model simulations.

The diffraction patterns ^{from the} crystal combined with ^a detailed knowledge ^{of the}

chemical structure make it possible ^to propose ^a model structure. By calculating ^{the structure}

Table 1.

A 1 Polymalone ^ester (main chain).

B 1 Polymalone ^ester (main chain/side group).

A2 Polytetrakis (pentyloxy) triphenylen alkandioat (monomer).

B2 Polytetrakis (pentyloxy) triphenylen alkandioat (polymer).

B 1 Main chain/side group system (polymalonester) .

A2 Triphenylene alkandioat (monomer).

B2 Polytriphylene ^derivate (polymer).

factors and accounting ^{for the} microscope ^transfer function, image simulations are performed

to test whether the proposed ^{structure and} calculated images ^{are related to the model.}

III. Electron diffraction results.

The diffraction pattern from the calamitic (A) and discotic (B) polymers ^{both in} crystalline (A 1, A2) ^and liquid crystalline (B 1, B2) phases ^{are shown in} figure 1 (a)-(d) .

In the case of the polymer ^{with the} rod-shaped azobenzene mesogenic unit, ^a polycrystalline fibre-diagram ^{was obtained} (Fig. la). ^This is because the sample ^{had been oriented} by

sgearing ^{and the a, b axes are} oriented around the c-axis. The repeat period ^of 1/35 Â1- along the c-axis corresponds approximately ^{to the distance between} mesogenic ^units along the chain. At least 8 high ^{orders are} present along the meridian and two reflections on

the equator plus ^two off-equator reflections ^on the hk2 layer line. The translational units

along ^{the chain are} clearly highly correlated, ^{as indicated} by ^the large number of maxima on

the meridian. Furthermore, ^a characteristic feature is ^a very intense 4th order maximum, corresponding ^{to a} repeat period ^of 8.7 A and indicating ^a sub-unit of this dimension along

the chain. It is not possible ^{to make a} definitive structure analysis ^{of the} poly-crystalline polymalonate. However, it is clear that the arrangements ^{of sub-units} along ^{the chains are} highly correlated. To obtain this polymer in the smectic phase ^after quenching, ^a side group with the same mesogenic ^{unit is} added, ^{in order to} prevent crystallization. ^{The immediate}

effect is a reduction in the long spacing ^from 35 A to 26 A and a dramatic loss of all higher-

order reflections. This is typical ^for liquid crystals (Fig. 1 b).

A similar analysis ^{of the discotic} material shows a loss of higher-order diffraction maxima in

passing ^{from the} crystalline (Fig. 1 c) ^to the discotic state (Fig. Id).

In order to understand the loss of higher ^order reflections, ^{it is} necessary ^to recall some

basic principles ^of diffraction theory. ^For X-ray scattering ^{it can} be shown that the intensity

distribution in reciprocal space i (q) ^{for a finite} crystal ^is given by [2] :

where

f ( q ) ^{is the structure} factor of the unit cell N is the number of unit cells

D (q) ^{is the} Debye/Waller ^factor accounting ^for thermal vibrations of the atoms

z (q ) ^{is the} reciprocal lattice factor describing the lattice statistics

s(q ) ^{is the} shape factor, taking ^account of the finite dimensions of the crystal

i (q )def scattering contribution due to isolated defects.

In principle ^the expression ^{for the} intensity distribution is the same for electron diffraction, provided ^the crystal ^{is thin} enough ^so that the kinematical (single scattering) approximation

holds and provided ^{the structure} factor is calculated as the Fourier transform of the electrostatic crystal potential.

From this general expression ^{it is clear that the 8 functions obtained from a perfect lattice}

are dampened ^at higher q-values by ^{the thermal} vibrations expressed by D ( q ) | 2 ^and

distortions in the lattice statistics expressed by z (q ). Since both the crystalline ^and liquid crystalline samples ^were investigated ^{in the solid state, we do not} expect D (q ) | 2 ^{but rather}

z(q ) ^{to be} responsible for the loss of higher order diffraction maxima.

In addition to the loss of higher orders, the diffraction spot ^{has a fine structure and each}

individual diffraction maximum is affected by ^the shape ^factor s(q). This may lead ^to

superstructure reflections which ^can only be resolved if the structures responsible ^{are small}

enough. If there is a distribution in size, ^the superstructure reflection will become smeared.

Crystalline main chain polymalonate. Smectic main chain/side group polymalonate.

Crystalline ^monomeric triphenylene. Discotic triphenylene.

Fig. ^1.

Diffraction patterns from smectic and discotic polymers ⁱⁿ crystalline (Al, A2) ^and liquid crystalline phases (B 1, B2).

Theoretical expressions regarding ^the exact nature in which disturbances in lattice statistics and the shape ^{factor are} responsible ^{for the observed} features in these experiments ^are

elucidated in the discussion. However, ^{it can be seen} immediately ^from equation (III.1 ) ^and figures ^{1-3 that the drastic loss in} higher ^electron diffraction maxima in passing ^{from the} crystalline ^{to the} liquid crystalline mesophase ^{for both smectic} and discotic samples ^indicates

disturbances in lattice statistics, i.e. the molecules are not situated ^on ideal lattice sites. Line

broadening ^{is observed} in the radial direction (r) (Fig. 2b). ^{A careful} analysis ^{of the} spot

contour in the smectic mesophases indicates fine structure arising ^{from the} shape ^factor s(q ) (Fig. 2c). ^A streak appears in the tangential ^direction (t), indicating loss in translational correlations in planes perpendicular ^{to the} streak, i.e. the smectic planes ^must be distorted in

some way. A theoretical derivation of this effect is given ^{in the} following ^section. Analysis ^of

Fig. ^2. - Analysis of diffraction spots obtained from main chain/side group polymalonate.

(a) Micrograph and diffraction pattern.

(b) Intensity distribution in radial direction (r).

(c) Magnification ^{of small} angle electron diffraction maximum and spot ^{contour lines.}

the diffraction spots in the discotic triphenylene ^{indicates some} broadening (Fig. 3b) ^{but a} superstructure reflection was not resolved in the contour plots (Fig. 3c). ^{It is} possible ^{that the}

statistical deviations responsible for loss of higher order maxima are distributed isotropically.

Finally, information about isolated defects is to be found in the continuous scattering background i (q)def-

IV. Effect of superstructure of electron diffraction pattern.

The wavefunction in the diffraction pattern ^{of an} object ^is the Fourier transform of the wavefunction at the exit face of the object. ^{Here, in order to obtain more} detailed information about the deformed structure from the diffraction pattern, ^{it is} necessary ^to derive ^an

expression ^{for the wave function at} the exit surface of ^a deformed crystal

where 03C8D is the wavefunction of the deformed crystal ^and 4ro ^{of the} undeformed crystal,

expressing the fact that the position of the molecule is displaced ^{from R to R + 8} (R).

Fig. ^3.

Analysis ^of diffraction spots- obtained from discotic triphenylene.

(a) Micrograph and diffraction pattern.

(b) Intensity distribution in radial direction (r).

(c) Spot-contour lines from small angle electron diffraction maximum.

The undeformed wave front is periodical ^{and can be} expressed ^{in a} Fourier series as

so that (IV.1 ) ^{now becomes}

If the displacement ^is periodical ^{in F, it can also be} expressed ^{as a Fourier series :}

so that

The amplitude ^{in the} diffraction pattern ^is given by the Fourier transform of 03C8D(R)

where h _{= q} + Q, ^i.e. reflections occur at q + Q ; ^each Bragg reflection at q has satellites at

positions given by q + Q.

If the deformation is continuous, ^the Bragg reflection has a continuous finite structure.

Therefore the form of the modulation can be directly derived from the fine structure of the

Bragg spots.

a) ^For small modulations :

Therefore (IV.3) ^becomes

If the modulation is expressed ^{as a} Fourier series :

so that

Therefore the amplitude _aQ of the satellite Q ^of reflection q ^is given by :

The relative intensity of the satellite Q ^{relative to the} intensity of its basic reflection

q gives ^{a direct measure} of the modulation bQ.

In the smectic material smeared satellites were observed (Fig. 2c) adjacent ^{to the}

diffraction maxima, indicating the presence of ^a superstructure. ^{The streak} clearly ^{shows that}

the translational correlation along the smectic planes ^is limited, ⁱⁿ agreement ^{with the} undulations observed in the micrographs. Work is in progress ^to resolve this fine structure.

V. High résolution imaging.

We have described in previous papers the difficulties in obtaining images from beam sensitive

samples ^{and the} optical criteria involved [11], [13-14].

It is necessary ^to choose spatial frequencies of interest by selecting ^a microscope ^transfer

function X (q ) which has ^a window in the appropriate frequency range. The required frequency range is selected by inspecting the diffraction patterns ^and may be different for every specimen. ^{For the} liquid crystal polymers ⁱⁿ question, ^the frequencies ^of interest all lie in the small angle range, where, under normal high resolution conditions, ^{there is no} phase

contrast transfer. For this reason, the contrast transfer function was calculated so as to give

maximum phase transfer and minimum amplitude ^{transfer at the} appropriate frequencies required ^to image 26 Â (smectic planes) ^and 18 Â (discotic columns) (Fig. 4). ^{It Ís - im-} mediately obvious that unusually large defocus values between 5 000 Â and 10 000 Â must be selected in order to image ^the required ^{features. For this reason} interpretation ^{of the} images ^is

not straight forward and in the following ^{we shall} (1) Compare images ^{of the} (unknown) ^{1. c.}

polymer ^{with the} (known) crystalline ^structure (2) Check that the evidence from electron

Fig. ^4.

Optimisation of electron microscope transfer function by defocusing ^{to obtain} required frequency ^transfer.

Fig. ^5a. - High resolution electron micrograph ^{of a} main chain liquid crystalline polymalonate ^{in the}

crystalline phase.

diffraction is in agreement ^{with the} images (3) Calculate the images which would be obtained from the proposed ^{model structure} taking ^account of the transfer function and see whether it agrees with the observed images (4) Calculate the wave function at the exit face of the sample,

insert the appropriate ^contrast transfer function in its Fourier Transform and calculate how the image is related to the potential distribution in the object.

The images ^and appropriate oriented diffraction patterns obtained for the liquid crystalline samples under consideration are shown in figures ^{2 and 3} (samples A2, B2). ^For comparison

the polycrystalline material with rod-like _mesogens (Al) ^{shows the} expected ^well ordered, straight ^lines corresponding ^{to the} repeat ^distance along ^{the chain} (Fig. 5a). It indicates a high degree of orientational and translational correlation along the chain. In ^some regions

characteristic defects arise, ^{such as the} simple edge dislocation indicated at D. The individual molecules are not observed under these imaging conditions but only ^the repeat period along

the chains. The reason for choosing these conditions is that it is this periodicity which will be

responsible ^{for the} planes observed in smectic materials.

Similarly ^the diffraction patterns ^{from the} crystalline triphenylene, ^{with its} sharp diffraction maxima and _many high orders indicates a crystallographically perfectly arranged hexagonal

lattice (Fig. 5b).

Fig. ^5b.

Resolution electron micrograph ^of crystalline triphenylene ^{monomer with} microdiffraction

pattern ^{in correct} orientation.

which must be recognized.

For these experiments ^the imaging conditions had been chosen such that information about the repeat period ^{in the smectic} planes and the discotic columns was transferred. Since the transfer function oscillates, ^it may transfer some higher frequencies either with the same or

opposite phase (Fig. 4). We have shown in ^a previous paper how to determine this [13]. ^The

Fig. ^6. - Liquid crystal polyester ^with mesogenic group in main- and side-chain, showing dislocation

with multiple Burger’s ^vector.

Fig. ^7.

High resolution electron micrograph of a discotic main chain polyester indicating chain tilt and vacancies.

purpose of the simulation calculations is ^{to ensure} that the appearance ^{of the} image ^after accounting ^for dynamical scattering and the transfer function is not drastically changed ^{and to}

determine how the potential distribution in the object ^{is related to the} image ^{as a} function of

object thickness and the defocus value.

in figure ^8a (polymalonate) ^and figure ^9a (triphenylene). ^This represents ^the crystalline

structure. We cannot determine the correct crystal ^{structure from the} experimentally

observed fibre pattern (Fig. la). However, the aim of this simulation is only ^to show whether the regularly spaced mesogenic ^units along ^the chain appear in the images in the form of a

single ^line.

The diffraction patterns calculated from these proposed structures are indicated in

figures ^{8b and 9b} respectively. The maxima along the meridian (in Fig. 8b) represent ^the diffraction intensities from the crystallographic repeat period along the chain. Experimentally

we observed fibre diffraction patterns ^{from the} polymalonate ^{Al so that the} spots ^{on the} layer

lines were reduced to arcs due to rotation and tilt about the c-axis.

However, ⁱⁿ comparing the simulations with experiment, ^{we shall be concerned} only ^with

the planes causing ^{the small} angle ^maxima along ^the equator. ^{While the} general features of the diffraction pattern ^are reproduced, ^the precise origin of the observed intense 4th order maximum is not yet ^clear.

For the crystalline discotic material B l, ^the agreement ^between experimental ^and

theoretical diffraction pattern ^{with its uneven} distribution of spot intensities according ^{to the}

Fig. 8. - Model of main chain polymalone ^ester single crystal (a) and calculated diffraction pattern (b).

Fig. ^9.

^{Model of} triphenylene single crystal (a) and calculated diffraction pattern (b).

different density ^{of atomic} occupation in the three principal directions _agrees well with the

experimental diffraction pattern.

Calculations of the images ^{from the} proposed crystalline structures are shown in figures ¹⁰

and 11.

Figure 10 shows the contrast change ^{in the} image ^{of the} polymalonate crystal ^{for a fixed}

thickness as a function of defocus. While the whole molecule is imaged ^at small defocus

values, ^at large defocus values only ^the repeat period along the chain is observed in the form of a single ^line.

Figure ^{11 shows the contrast} change ^{of the} triphenylene crystal ^{for a} fixed thickness as a

function of defocus. Again, while the whole molecule is imaged ^at small defocus values, only

the hexagonal arrangement is retained at large ^defocus values, ^{but the} general triangular ^form

of the molecule is retained.

Fig. ^10. - Simulated images of main chain polymalonate ^{as a} function of defocus value Af .

Fig. ^11.

Simulated images ^of triphenylene crystal ^{as a} function of defocus value Af ’.

Therefore, ^{in the} crystalline ^{state where} many diffraction orders are observed, it would be

possible ^to image the whole molecule. However, ^the specific aim of this work was to obtain

images ^{from the} liquid crystalline materials A2 and B2 in the glassy ^{state. Here} the electron diffraction patterns ^{showed a} dramatic loss of all the wide angle reflections.

Therefore it is not possible ^to obtain information about details of the molecular arrangements from these samples ^but only ^about periodicities in the small angle range (i.e.

the smectic planes and discotic columns) ^{which can} only ^be imaged ^at large ^{defocus values. It}

can be seen immediately ⁱⁿ figures ¹⁰ (d) ^{and 11} (d) ^{that the} agreement ^between experimental

results and the simulations is excellent.

The experiments ^on liquid crystalline polymers ^{and the} simulations have therefore shown that there is a one-to-one correspondence between the structure of the object (as ^determined by ^{its chemical nature} and the electron diffraction pattern) ^{and the} high resolution images.

This result can only ^be understood from a theoretical stand-point, ^even beyond ^{the weak} phase approximation. ^The proof ^{will be} given ^{in the next section.}

VII. Theoretical basis for the interprétation ^{of the images.}

Our results indicate that the postulated structural models for the disc-shaped triphenylene ^and

the linear polymalonate molecules, ^{based on electron} diffraction and phase ^contrast imaging,

agree with the simulated images of these molecules. In order to understand this result and to

determine whether a one-to-one correspondence between the projected potential ^{of the}

object ^{and the} image ^{can be} expected theoretically, ^the following considerations have been

formulated :

Consider the electrostatic potential ^{of a} perfect crystal ^{as the} superposition ^{of the} potentials

of the constituting ^{atoms :}

where Vi ^{is the} potential ^{of atom i at the site} rl.

The projected potential ^{is then :}

where R, Ri ^{are the} projections ^{of r,} r 1. V p, ^{i is the} projected potential ^{of atom}

i.

The model now assumes that perfect layers ^{of the} crystal ^{in the x,} y plane, ^{one unit cell} thick, ^are shifted with respect ^{to the} layers ^{below or} above, causing ^a disturbance in the z-

direction, which is also the beam direction. The atom in layer n ^{is now} displaced by ^{a vector}

n.

The projected potential ⁱⁿ (VII.2) then becomes :

where the summation over i extends over the atoms in one layer. ^{If the} projected potential ^of

one atom is spatially arranged ^over the different layers :

In the case of perfect correlation along z, all atoms of one type overlap, yielding ^{a very} large projected potential peak ^{at the centre of the} projected ^{atom. When} perfect correlation in the z-direction is disturbed, ^the projected potential ^{is less} sharply peaked.

If the number of layers ^is sufficiently large, (VII.4) ^{can be} approximated by ^an integral :

This can be expressed ^{as a} convolution product :

The Fourier transform f gives ^the scattering factor of the average projected ^atom

where p (q ) ^is the Fourier transform of P ( R) and q ^{is the} scattering ^{vector in the} plane ^of projection. If P (R) ^is Gaussian, p (q) ^{is also} Gaussian, ^{so that} clearly ^the higher ^{order Fourier}

components ^{of the} scattering ^{function are} dampened.

The total projected potential ^{is now :}

A the electron wave length

E the accelerating potential

d specimen ^thickness V(R) crystal potential

In order to estimate oVp(,R) ^the following procedure ^is adopted : ^{since the atoms do not} superimpose exactly along ^{the beam} direction, ^the average projected potential ^{can be}

considered ^{as a} smeared-out potential ^{of the atoms. The} average potential ^{of one atom can be}

estimated as follows : consider ^an atom as a sphere ^{with a radius R and a} charge

Ze in the centre, ^{where Z} is the atomic number. The electrostatic potential averaged ^{over the}

whole sphere ^{is then}

where the surface of the sphere ^{is at zero} potential. ^In practice, ^the average potential ^is

reduced due to screening ^{of the} electrons, ^{so that} we assume :

Projecting along ^the object thickness d then yields :

and finally :

The weak phase object approximation ^{holds if}

so that

one obtains

with d expressed ⁱⁿ Á.

Therefore the weak phase approximation holds for light elements, ^{such as} those under

consideration, ^for relatively large thicknesses ^as compared ^to crystals. ^{It can} thus be used as a

basis for discussing the characteristics of the high resolution images ^{in the} samples ^under

consideration. Furthermore, since the projected potential (Vp) ^{varies over} relatively large distances, the column approximation holds, ^so that the wavefront at the exit face of the object

still has ^a one-to-one correspondence ^{with the} projected ^{structure of the} object. ^{This one-to-}

one correspondence ^is kept ^{in the} high resolution images provided the defocus distance is not too large.

If absorption ^{occurs, it can} be introduced using ^an absorption ^{function IL} (R) ^{so that}

The ^wave function in the diffraction plane ^is given by the Fourier transform of (VII.19)

where

and

This complex ^{function must be} multiplied by ^the phase transfer function of the electron

microscope

where

and Cs ^{is the} spherical aberration of the lens. Of is the defocus value. From (VII.20) ^{the wave}

function is the diffraction plane ^becomes

The imaginary ^{terms in} (VII.23) become second order in the final intensity ^{and can be} neglected. ^Therefore

For the relevant spatial frequencies ^{in the} specimens ^under discussion, q ^{is so} small that the

term q 4 ⁱⁿ (VII.22) ^is negligible, ^{so that}

Fourier transforming (VII.27) ^then yields ^{the wave} function in the image plane which, using

the Poisson equation

can then be written as [17] :

This expression ^{holds even if the} w.p.o. approximation ^{is not} valid but the _p.o. approximation

holds [18]. ^{It can be seen from} (VII.29) ^{that a zero defocus} object ^shows only absorption

contrast. On defocusing, ^the image reveals the projected charge density //p(R)> ^{of the}

object.

(b) Large defocus ^values.

For large ^defocus values, the transfer function was adjusted by defocusing ^{such that} Sin y f--- - ^{1 and} Cos X = 0 for all relevant values. This focus is called Scherzer or optimum

focus.

Equation (VII.24) ^{now becomes :}

The image intensity ^is given by ^Fourier transforming (VII.30) ^and taking ^the square ^so that,

up ^to first order :

In this case, which corresponds ^{to the} experimental conditions under discussion, ^the image

contrast is proportional ^{to the} projected potential.

VIII. Image restoration.

Two major problems arising ⁱⁿ high resolution electron microscopy ^{of beam sensitive} samples

are poor ^contrast and _poor signal/noise ^{ratio. Since we are} interested in the analysis ^of

molecular defects and statistical deviations from lattice positions, ^{this is} clearly ^a major handicap.

In order to assess the quality ^{of an electron} micrograph, ^both light diffractometry ^{and the} digital image analysis ^{were used. The} diffraction pattern ^{from the} image immediately gives ^a

first indication as to whether a micrograph is suitable for further analysis. ^{In our} experience

with these samples, Fourier space frequency filtering ^{delivered the best results, but many}

alternative methods ^are available.

However, in order to avoid misinterpretation, ^{it is} important ^to understand what is

happening ^{to the} image [20-21]. ^The digitized image I(x,y) ^can be written (Fig. 12)

where * denotes convolution.

The following quantities ^{are related to the} micrograph : M(x, y ) ^motif of the unit cell

L (x, y ) lattice function

B (x, y ) ^noise.

The remaining quantities ^{are related to the} sampling procédure :

LS (x, y ) sampling ^lattice function, consisting of delta functions at the sampling ^lattice points F (x, y ) ^window function, selecting ^{the area used for} digitizing

Rs (x, y) the function representing ^the sampling aperture ^{at each} sampling point.

Denoting the Fourier transform of functions in real space (represented by ^{upper case} letters) by functions in reciprocal space in lower case letters the Fourier transform of the

sampled image ^is given by

To reduce noise, ^a filtering aperture ^{a is} introduced in reciprocal ^{space so that}

Fig. ^12.

^Schematic representation ^of digitisation procedure for discotic triphenylene ^molecule.

the reciprocal lattice. A theoretical treatment of image processing ⁱⁿ non-periodic ^structures

has been discussed in detail elsewhere [22]. ^{It was our aim to find} adequate sampling parameters ^{and an} aperture large enough ^{to retain} information about the positions ^of

molecules which were not situated precisely ^on the lattice. The result of such an analysis ^is

shown in figure ^{13 for the} polymalonate ^and figure ¹⁴ for the discotic triphenylene. ^On figure ^{13 the} power spectrum is indicated in the top ^{left hand corner} of each restored image.

The size of the aperture used is marked by ^a circle. It is clear that both the aperture ^{size and} shape ^{must be} carefully selected and that aperiodic information is _very difficult to restore, ^as

we have discussed previously [13]. Figure 13 shows that the application of a circular aperture is unsuccessful in restoring aperiodic ^linear structures, ^{no matter} which aperture ^{is used.}

However, in the case of isotropic structures the Bragg aperture works well (Fig. 14) ^{and in}

some cases even the triangular shape ^{of the discotic molecule can} be discerned after image restoration, ^{as is to be} expected from the simulations. This indicates that regions ^are present

Fig. ^13. - Change ⁱⁿ superstructure ^of reconstructed image ^and apparent ^defect density ^{as a function}

of aperture ^size.

(Power spectrum ^and aperture ^{size are} indicated in top left hand corner.)

Fig. ^14.

Frequency ^filtered images from discotic triphenylene ^columns relating aperture ^size (r) ^to periodic ^structure (p ).

(a) ^First order, r

1 /p.

(b) Second order (negative phase), r

4/p.

(c) ^{First +} second order (negative).

(d) ^First order, r

2/p.

(e) Second order (negative phase), r

2 /p.

(f) ^{First +} second order (negative).

in the discotic samples without distortion with respect ^{to the} perfect crystal. ^In figure ^{14 the}

radius of the aperture ( r) ^{is related to the} periodicity (p ) . The best results were obtained when the second order was transferred with negative phase, suggesting ^{that our true defocus}

value did not correspond exactly ^to the calculated ^one (Fig. 4).

IX. Conclusion.

For the specimens ^under consideration, ⁱⁿ which (V p) ^{varies slowly, the} wave-function at the exit face of the object ^{as well as the} high resolution images ^{show a} one-to-one correspondence

with the projected ^{structure of the} object. ^{On this} basis, ^{we can treat} the observed images ^as

real structures and formulate the following conclusions about the structure of liquid crystalline smectic and discotic polymers ^{in the} glassy ^{state :}

(1) Electron diffraction from perfect crystals ^and liquid crystals ^{in the} glassy ^{state indicate a}

dramatic loss of higher order reflections for the latter.

disturbances are isotropically arranged ^and their effects on lattice spacings ^are averaged ^{out in}

-normal selected area electron diffraction. They ^are responsible for the loss of higher ^order

diffraction maxima. The disturbances can be analysed, however, from selected area

diffractograms ^{from small areas in the} images.

(5) Lattice defects in the form of single ^and multiple dislocations are shown to occur in the smectic planes.

(6) From the diffraction evidence combined with the images, ^and knowledge ^{of the}

chemical structure, models for the ideal structures are proposed. These models are used as a

basis for the image simulations. A perfect correspondence is found between model and simulated structures.

(7) The excellent agreement between the experiment and simulations can be understood

only if the weak phase approximation holds. A theoretical model is proposed in which the interaction between the electron beam and the sample is calculated. The theory predicts ^weak phase behaviour for these specific samples and under the imaging conditions employed ^even

for relatively ^thick samples.

(8) ^{The method of} image restoration using spatial frequency filtering ⁱⁿ reciprocal ^{space is}

shown to give satisfactory results for the discotic samples ^{but not} for the smectic planes. ^The

reasons for the differences are discussed.

From a structural point ^of view, these results indicate that relatively small deviations from ideal lattice positions ^occur during transition from the crystalline phase ^{into the} liquid crystalline meso-phase. ^In the smectic structures, ^the planes causing ^{the small} angle

diffraction maxima arise from the repeat period along the chain. In order to form ondulations,

translational motion parallel ^to the chains is required. Since the Burger’s ^{vector in the}

observed dislocations is also parallel ^{to the chain} direction, relatively easy motion along ^the

chains can be expected. Some dislocations are shown to have Burger’s ^{vectors which are} multiples ^{of a lattice} spacing.

For the discotic material, ^the hexagonal arrangement of the columns is disturbed by ^small displacements of the discs in the individual _xy planes causing ^a shift of the projected ^structure

in the z-direction.

Acknowledgement.

We gratefully acknowledge the Deutsche Forschungsgemeinschaft ^{for financial} support

within the framework of the Sonderforschungsbereiche ^262.

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