An investigation of the spot profiles in transmission electron diffraction from Langmuir-Blodgett films of aliphatic chain compounds

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Submitted on 1 Jan 1990

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An investigation of the spot profiles in transmission

electron diffraction from Langmuir-Blodgett films of

aliphatic chain compounds

I.R. Peterson, R. Steitz, H. Krug, I. Voigt-Martin

To cite this version:

An

_{investigation}

of the

_{spot profiles}

in transmission electron

diffraction from

_{Langmuir-Blodgett}

films of

_aliphatic

chain

compounds

I. R.

_Peterson,

R.

_Steitz,

H.

_Krug

and I.

_Voigt-Martin

Institut für

_{Physikalische}

Chemie, Johannes

_Gutenberg

Universität, Jakob Welder

Weg.

11,

D6500 _Mainz, F.R.G.

(Reçu

le 13

_septembre

1989, révisé le 5 _janvier 1990,

accepté

le 17 _janvier

₁₉₉₀₎

Résumé. 2014 Les

_profils

des réflexions de

_Bragg

_provenant d’une couche de

Langmuir-Blodgett

sont dérivés selon trois modèles

_plausibles

de

_{l’organisation}

moléculaire : une

_phase

_{polycristalline}

à ordre orientationnel de

_longue

_portée;

un

_paracristal

possédant

une densité de défauts

ponctuels qui

ne

_dérangent

_pas les rangs moléculaires ; et une

_phase

_{smectique hexatique}

du _type

Nelson et

_Halperin

dans la limite «

_{Debye-Hückel »}

où les interactions entre les dislocations sont

faibles. Les trois

_prévisions

sont nettement différentes l’une de l’autre. Des résultats

expérimen-taux à

température

ambiante sont

_présentés

_pour des couches de

Langmuir-Blodgett

fabriquées

à

partir

d’un

_lipide,

le DMPE, et du sel de cadmium d’un acide gras insaturé, le

22-tricosénoïque.

Seule la théorie de Nelson et

_Halperin

donne un accord _{satisfaisant,}

malgré

des deviations

_qu’on

peut attribuer à des défauts du _type

_{paracristallin}

ou à un _comportement non

_ergodique

de la

couche. On en déduit des densités de dislocations de l’ordre de 10-4 à 10-5 par molécule. Abstract. 2014 The

profiles

of the transmission diffraction _spots from a

_{Langmuir-Blodgett}

film of

aliphatic

chain

_compound

are derived from three

_plausible

models of molecular

_{organisation :}

a

polycrystalline phase

with

_long-range

orientational order; a

_{paracrystal possessing}

a

_density

of

point

defects which do not

_interrupt

the lattice rows ; and a Nelson and

_Halperin

hexatic smectic

phase

in the

«Debye-Hückel»

limit of

weakly-interacting

dislocations. The three

resulting

predictions

are

_distinctly

different.

_Experimental

results are

_presented

for the _{room-temperature}

diffraction patterns from

Langmuir-Blodgett

films of a

_lipid,

_DMPE, and the cadmium soap of a

fatty

acid, 22-tricosenoic.

_Only

the Nelson and

_{Halperin theory gives}

a

_satisfactory

fit,

although

there are deviations attributable to

_{paracrystal-type}

defects or to

_non-ergodic

behaviour of the

layer.

Dislocation densities of between 10-4 and 10-5 per molecule are deduced. Classification

Physics

Abstracts 61.70 - 68.55 - 61.14

- 61.50K - 07.80 - 05.50 - 03.40

1. Introduction.

Electron diffraction has been used since the 1930s

_[1,

_2]

to

_investigate

the molecular

_packing

on the nanometre scale in the

_organic

films

_{deposited by}

the

_{Langmuir-Blodgett (LB)}

technique.

The diffraction

_patterns

obtained contain a number of discrete

_spots,

which in

these and

_subsequent

studies

_[3-7]

have been

_interpreted

in terms of a

crystalline packing

of

the molecules.

While the

_{crystalline hypothesis}

has the

_advantage

that many such materials are known ànd

that the

theory

of their diffraction

_patterns

is well

understood,

it has never been

_completely

satisfying

as an

explanation

of LB film diffraction results. Ideal

crystals

_{produce sharp}

diffraction

_spots,

even in the _presence of thermal disorder : the effect of an

_equilibrium

distribution of

_phonons

is to reduce the

_{peak intensity (by}

the

_{Debye-Waller factor)}

and to introduce a smooth

_{scattering background,}

without any

peak broadening [8].

In contrast, the reflections from LB films

_{generally display peak}

_broadening

as well as

_arcing,

a fact which was

first

_brought

to attention

_by

Fischer and Sackmann

_[9],

and

_{subsequently by}

Garoff et al.

[10].

In the usual case of a

_polycrystal,

the distribution of

_grain

orientations is

_isotropic,

and the diffraction

_pattern

consists of

_{Debye-Scherrer rings.}

In contrast, reflection

_high-energy

electron diffraction studies of LB films

_typically

show a fibre

_pattern.

This is

_normally

considered to arise from

_{polycrystalline}

_aggregates

with statistical fluctuations about a

preferred

molecular orientation

_(the

fibre

_{axis) [11].}

An additional feature of the diffraction

_patterns

from LB films is that the number of reflections is much smaller than that obtained from

_{single crystals}

of

_molecularly

similar materials

_[11].

The

_missing

diffraction

_spots

are those

_{of high}

_{order. This may also be ascribed} to fluctuations in molecular orientation or

_position,

and indicates that their correlation

_length

is small.

The observation of localised

peaks

of diffraction

_intensity

in

_spite

of these fluctuations of molecular

_positions

and orientations indicates the _presence of orientational order of the lattice

_extending

over much

_larger

distances. Information about this can be obtained more

conveniently using polarised microscopy.

In recent studies of LB films of the

_fatty

acids and

soaps

[12],

textures

_typical

of hexatic smectic

_mesophases

have been

_reported.

These are

characterised

by

a continuous variation of the

_{preferred optical}

axes at almost all

_points.

At

some

_points,

the orientation of the

_optical

axes

_changes

much more

_rapidly

than

_elsewhere,

forming

a

_pattern

of lines similar in some

_respects

to

_grain

boundaries. However

_they

do not enclose « domains » of constant

_orientation,

and most are _open

_(i.e.,

have two

_ends),

without

Y-junctions.

The

change

of orientation across these lines is

_equal

to 60° within

_experimental

error, and the

_resulting

diffraction

patterns

which can be indexed as

arising

from

grains

differing

in orientations

_by

60° have been ascribed to «

_hexagonal

_{twinning »}

_[13].

The ends of the lines of

_{discontinuity}

have been identified as the orientational defect structures known as

disclinations

_[14-16].

Taken

_together,

the observed details of the texture

_imply

_complete

loss of translational correlation within

_{optically-resolved separations}

in all directions within the

monolayer plane (as

in a

_liquid),

while correlations of the lattice orientation are retained to much

_greater

distances,

typically

tens to hundreds of ktm.

It is of more than academic interest to understand the

_physical

basis for the differences between

_crystalline

and smectic

_behaviour,

because in

_attempts

to

_apply

LB films in

_practical

applications,

characteristic smectic textures _appear to be

_responsible

for many of the

departures

from desired behaviour

_[17,

_18].

Moreover,

one of the

_{commonly expressed}

aims

of LB research is « molecular electronics »

_[19],

where the local environment of each and

every molecule is

expected

to be

_important.

Since the diffraction

_pattern

is

_readily

accessible and

_{produces distinctly}

different results with

_crystalline

and smectic

materials,

it seems reasonable to test theories of molecular order in these

_{phases against}

their

_predictions

of the diffraction

pattern.

It is well known that the interaction of electrons with matter is very

strong,

and the

possibility

of

_multiple

diffraction

_{(or dynamical scattering)}

must be very

seriously

considered

in any detailed correlation of a structure and the

_resulting

diffraction

_pattern.

However Dorset has shown

_[20]

that the

_{single-scattering (kinematic) approximation}

holds well for

polymers

it has been shown both

_{experimentally [21] and theoretically [22]}

that the kinematic

approximation

is valid for

_sample

thicknesses of a few tens of nm. Hence for the case of

organic monolayers

thinner than 3 nm

_supported

on an

amorphous

substrate less than 20 nm

thick the kinematic

_theory

is

_{quite adequate.}

In the

_present

work,

three

_possible

models of molecular

_organisation

are

_{analysed using}

the kinematic

approximation.

Each is

_plausible,

and has at some time been

proposed seriously

to describe the order in smectics or LB films. The first is a

polycrystalline

texture, in which

regions

of defect-free

crystal

with constant orientation

_(«

_{grains »)}

are

_{separated by sharp}

boundaries. This is the model

_{implied by}

the normal nomenclature used in diffraction

studies,

which was

_developed

for

_inorganic

materials. In order to make this model conform with the observed

_long-range

orientational order in LB

_films,

it is necessary to assume

that,

as a result of some

_mechanism,

the orientations of

_{neighbouring grains}

are

_highly

correlated.

The second is the Nelson and

_Halperin

_{(NH) theory}

of hexatic smectic

_mesophases,

which has in

previous

papers

[14,

15,

23]

been

proposed

as a model for

fatty

acid LB

_{organisation.}

Unfortunately, although

Nelson and

_Halperin

have

_{extensively analysed}

the

expected

elastic and

_{thermodynamic properties}

of these

_{phases [24-27],}

and have derived functions related to

scattering,

nowhere in their work do

_{they directly}

derive the diffraction

pattern.

This

oversight

is rectified in the

present

paper for the

physically plausible

«

_{Debye-Hückel »}

limit of

_{weakly interacting}

dislocations.

The third model is a

_{paracrystalline}

model of disorder

_[28].

_{Historically,}

the

_paracrystal

concept

proposed by

Hosemann and

_Bagchi

was the first

attempt

to

_explain

the diffraction

spot

broadening

encountered with _many

_organic

_materials,

and was

_developed

from

_plausible

models of lattice disorder in one dimension

_{[29], assuming}

_that,

unlike the case of

_inorganic

materials

[30],

dislocations can be

_{ignored. Unfortunately}

for this

_theory,

dislocations have

recently

been observed

_directly

in

_organic

thin films

_[21],

and there is no theoretical

justification

for « Gaussian error

propagation »

in two or more dimensions. The

_original

formulation has been

_ably

and

_definitively

criticised

by

Bramer and Ruland

[31],

and

_by

Crist and Cohen

_{[32]. Although}

there are sufficient free

_parameters

in the

_theory

to fit the variation of

_spot

widths observed from _many

_organic

_materials,

_{they pointed}

out that the

_predicted

small-angle scattering

does not

_correspond

to

_experiment.

Hence there are no

_grounds

for

believing

that the

_parameters

extracted for a

_particular

material

_provide

any

insight

into its

local molecular

_{organisation.}

Although

Hosemann’s version of

_{paracrystal theory}

can therefore no

_longer

be taken at face

value,

it does address a real

_phenomenon

which is

_ignored

in the NH

_theory.

Hosemann considered line

_broadening

to be due to defects such as

_vacancies,

chain

_kinks,

_impurities,

random

_crosslinks,

or

_regions

of different but near-commensurate molecular

_packing,

whose overall

_Burgers

vector is zero, and it is clear that defects of this sort exist. The NH model treats

_only

the effects due to dislocations.

The methods used in the NH

_theory

indicate how the

_{underlying physical assumptions}

of the Hosemann model

_might

be treated

_correctly

in two dimensions. The defective

_assumption

of « Gaussian error

_{propagation »}

must be

_{replaced by}

the self-évident one of mechanical

equilibrium.

The

_{resulting analysis}

of the modified model is

_presented

here. As in the NH

model,

the lattice is assumed to be

_{mechanically isotropic,}

an

assumption

which is exact for

hcp packing.

Finally,

the diffraction

_patterns

_{predicted by}

each of these models have been

_compared

to

experimental

results. Two

_{monolayer-forming}

substances were

_{investigated,}

in case a

particular

substance should turn out not to be

_typical.

For this reason

_also,

an

_attempt

was

made

to select substances from different

_{categories, although}

the selection could not be

hexagonal

close

_packed.

Two substances were chosen :

_firstly,

a

_{biological lipid,}

dimyristôyl

phosphatidyl

ethanolamine

_{(DMPE) ;}

and

secondly,

the cadmium soap of a

_long-chain

unsaturated

fatty

acid,

22-tricosenoic acid

_(Cd

_22TA).

Molecules of both substances are

similar in

_possessing

_{long aliphatic}

chains as an

_important

structural

feature,

although

the number of chains per molecule and the details of the

hydrophilic

headgroups

are

_quite

different.

Provided that the

_predictions

of the three models differ

_by

more than the

_experimental

error, then

agreement

of one of the models with the measured value can be taken as a

criterion of

_{validity. Conversely,}

if the

_prediction

of a

_given

model

_disagrees

with the measured value

_by

more than the

_experimental

error, then this can be taken as

_grounds

for

rejecting

it. However in this case it must also be considered whether the model can be

plausibly

modified to

_bring

its

_predictions

into line with observation.

2.

_{Experimental.}

The LB

_deposition

was

_{performed using}

either a Lauda

_type

FW-1

_{analog-controlled trough}

or a Nima

_type

TKB 2410 A

_{computer-controlled trough.}

The

_subphase

water was

_purified

using

a

_{Milli-Q recycling deioniser/active charcoal/filter}

unit fed with distilled water. For the

deposition

of cadmium

tricosenoate,

1.4 x

10- 4

M

CdCl2

and

10- 4

M

_NaHC03,

both Merck

p.a.

grade,

were added.

_Spreading

solutions of both DMPE and 22-tricosenoic acid of

approximately

1

_g/1

concentration were

_prepared

in

_{freshly purchased}

_p.a.

_grade

chloroform.

The 22-tricosenoic acid was

_synthesised

as described

_{previously [33]}

and was found

_using

NMR to contain 1.5 % of the 21-isomer. The DMPE was

_purchased

from Merck and was used

without further

_{purification.}

Hydrophilic

Formvar-coated electron

_{microscope grids}

were

_{prepared using}

a

_technique

described

_{previously [34, 35]. Using}

an

_optical

interference

_technique

described elsewhere

[36]

the thickness of the Formvar film was determined to be

_typically

less than 20 nm. These were coated with

_monolayers

of DMPE at a surface _pressure of 27 or 39

_mN/m,

or with

cadmium tricosenoate at a surface pressure of 35

mN/m,

both at a withdrawal

speed

of

83

_pm/s

and

temperature

of 20 ° C.

It is well-known that radiation

_damage

_poses a serious

_problem

in electron

microscopic

studies of materials

_{consisting largely}

of

_aliphatic

chains

[37-39].

Beam

_damage

leads

initially

to

_spot

_broadening

_and,

in extreme cases, the lattice

expands measurably, inducing

a

_phase

change

from an orthorhombic to a

_hexagonal

subcell

_{packing [40, 41].}

For this reason the

electron diffraction

_patterns

were all obtained under low-dose conditions.

_Changes

in the diffraction

_patterns

were detectable after four times the standard _exposure.

Transmission electron diffraction

patterns

from the Formvar-coated

_grids

were obtained in a

_Philips

EM300 electron

_microscope

at normal beam incidence and calibrated

_against

an

evaporated

thallium

_(I)

chloride

_sample.

The

accelerating voltage

in all cases was 80 kV. The

sample

area

_contributing

to the diffraction

pattern

corresponded

to one hole of an electron

microscope grid,

i.e.

_{approximately}

100 pm in diameter. The

images

were recorded on Ilford

PANF film and

_{developed using}

Kodak D19

_developer

for 5 min. Two-dimensional scans of

optical density

of the

_{resulting negative}

were obtained

using

a Plumbicon camera whose

output

was

_digitised

and stored in

_computer

_memory as a 512 x 512 _array of 8-bit

brightness

values.

The coordinates of the undeviated beam centre were determined

by averaging

those of the

six

₍₁₀₀₎

diffraction

_peaks.

The small linear shift in baseline caused

_by

nonuniform illumination of the

_negative

was

_subtracted,

as was the

circularly-symmetrical scattering

-All

_{processing subsequent}

to

_background

subtraction involved

_{least-squares}

fitting

to theoretical

_{peak profiles.}

Almost all

previous

studies have

_investigated

the fit to the

one-dimensional Lorentz

_profile

₍₁

+

ar2)-1.

However in

_appendix

D it is shown that in two

dimensions,

exponentially decaying

correlations lead to a

₍₁

+

ar 2)- 1.5

behaviour. A variable

exponent

was therefore included in the program, with the

_advantage

that the Gaussian case is

given by

the limit of

_large

_exponent.

To obtain a contour

_plot,

the

_{digitised intensity}

values cannot be used

_directly

because

_they

are not defined at a continuum of

_points.

To derive the

_plot

of

figure

_2,

the

background-corrected values were convolved with a

smoothing

function

_{S(x, y)}

defined

by :

when

otherwise,

where the coordinates x

_{and y}

are

_expressed

in

_pixel

units. The

_resulting

intensities at the

pixel

centres differ

_minimally

from the

_original

measurements, but in addition are defined at all intermediate

_points

with continuous first

derivatives,

so that the directions of the contour lines _vary

_smoothly.

The smoothed function was not _{used in any of the model}

_fitting

procedures.

The

_histograms

of

_figure

3 were obtained

by

summing

normalised Gaussians for each

peak

measured

₍₁₈

in each

_case)

with a mean

equal

to the observed axis ratio and a standard deviation

_equal

to that for the measurement. To determine the

_aspect

ratio of a

_peak,

all the

pixel

values

_{immediately surrounding}

_{it in the range from 20 %} to 100 %

_{peak height}

were

least-squares

fitted to the theoretical function with the intermediate value 5 for the

exponent.

To determine its standard

_deviation,

the variances due to Gaussian fluctuations in each

_pixel

were

_added,

_{assuming essentially}

linear

_dependence

on each of the

_pixel

values. The standard

deviation of each

_pixel

was

_{conservatively}

taken

_equal

to the entire rms difference between

the measured values and the best fit

_{function, i.e.,}

_assuming

no

_systematic

error.

To determine which

_profile

was a better fit to the

₍₁₀₀₎

reflections

_{(Fig. 5),}

the

least-squares fit routine considered all

pixel

values within a _{square of side}

equal

to three times the

typical

full-width half-maximum

_(FWHM).

The

_{goodness-of-fit}

criterion was taken to be the value of rms error. Due to _{the low-dose exposure, the silver}

_density

in the film was small. No evidence _{for saturation of the film response} was

_observed,

and in consequence no

compensation

for

_nonlinearity

was

_applied.

It should be noted that any residual

nonlinearity

will tend to favour the Lorentzian fit relative to the Gaussian.

In tables 1 and

_II,

each value of radial

_peak

width FWHM

_given

is the mean of the six values

FWHM,

from the

_{corresponding sample.}

The value

_quoted

for its standard déviation was

calculated from the N = 6 values

using

the usual statistical formula :

3. Results.

3.1 THEORETICAL. - The theoretical

_spot

_profiles

for the

_{polycrystalline, Nelson-Halperin}

(NH)

and

_paracrystal

models are derived in

_appendices

A,

B and C

_{respectively.}

The

predictions

can be classified into three distinct

_aspects,

_concerning

the

_aspect

ratio of the

_spot

contour, the

profile

of radial sections

_through

the

_spot,

and the

_relationship

between the

_spot

widths of different diffraction

order,

respectively.

Contour

_Aspect

Ratio. The most

_readily

measurable feature of the

_spot

contour, or locus of

intensity.

The

_{polycrystalline}

model

_gives

1,

the NH model

_predicts

B/3,

while the

_paracrystal

model can

_give

values

_larger

or smaller than

unity depending

on the balance between

_isotropic

and

_anisotropic

defects in the lattice.

Profile.

The radial _spot

_profile

is Gaussian for the NH

model,

Lorentzian for the

polycrystal,

and power law for the

paracrystal.

Spot

Width Variation.

_Although

all three lead to

_spot

_broadening,

the FWHM

_(full-width

half-maximum)

spot

width is

_independent

of diffraction order in the

_{polycrystalline}

case and

proportional

to diffraction order in the NH case. In the

_paracrystal

case there is a critical distance from the

_origin,

inside of which the

_spot

FWHM is zero, and outside of which there

are no _spots at all.

Hence all three

_predictions

are well defined and

_distinctly

different,

an ideal situation for an

experimental

test.

3.2 EXPERIMENTAL RESULTS. - TED

_negatives

were obtained from 3 DMPE

samples

and

3 Cd 22TA

_{samples. Figures}

1 shows all the diffraction

_patterns

of the two

_types.

When measured

_{conventionally by optical comparison}

with a

graduated

scale the

d-spacings

of all

first-order reflections were the same and

_equal

to 0.416 ± 0.008 nm,

corresponding

to

_hcp

packing

with unit cell vector

_equal

to 0.481 ±0.010nm. It can be seen that the first-order

(100)

spots

are oval in

_shape,

with

tangential

and radial

symmetry,

and

_tangential

dimensions somewhat

_larger

than radial. Faint

(110)

spots

are

_visible,

but

_higher

order diffraction

_spots

are

_missing.

Fig.

1. -

Transmission electron diffraction

pictures

from all of the

_{samples :}

Cd 22-TA :

(a)

55879

(b)

The

_negatives

were

digitised

and stored on diskette as a file of 512 x 512

bytes

for further

analysis.

The contour

_plot

of

_figure

2 was obtained from one of the

₍₁₀₀₎

_peaks

of the

figure

la

_negative

as described in section 2.

Monolayers

on _{Formvar substrates of the cadmium soaps of the}

fatty

acids

_[42]

and DMPE

[43]

have been

_{previously reported}

to

_{display hcp}

molecular

_packing.

However,

just

as

reported

in reference

[44],

the intensities of the six

₍₁₀₀₎

_spots

were found to fluctuate

_by

more than 20

%,

their radial and

_tangential

widths

_by

more than 10 % and their

d-spacings by

2 %. These fluctuations are

_greater

than the maximum

_probable

error in the measurements.

Fig.

2. - Contour

plot

of one of the

₍₁₀₀₎

_spots of the DMPE diffraction

picture

shown in

_figure

1, after

digitisation, background

subtraction, and convolution. The undeflected beam is in the direction of the bottom of the page.

Figure

3 shows

_histograms

_plotted

for the

(100)

and

₍₁₁₀₎

_peaks

of both DMPE and Cd 22-TA

_samples

as described in section 2. In each

_category

there are three

_samples

with six

spots

each,

giving

a total of

_{eighteen peaks}

for each

_histogram.

The values observed are marked

_by

symbols plotted

on the curves, with different

symbols indicating

different

samples.

For the

(100)

spots,

the values are clustered near

B/3,

the lowest values

_being

within

_experimental

error of

J3

but the

_highest

values

_{significantly larger.}

The

_spread

of values for the

peaks

from the one

_sample

was

_considerably

smaller than the

_spread

between

_samples.

Fitting

to

the

_{(110) peaks}

was

_difficult,

_probably

related to the fact that in the best case the

peak

was

_only

a factor of 5

_higher

than the

_noise,

and two of the DMPE

_{(110) peaks}

could not be

_{distinguished}

from noise. When the best-fit

_algorithm

was allowed

_{simultaneously}

to _vary the

peak

coordinates,

amplitude,

and

_widths,

the result in 80 % of cases was very slow convergence to a

_{completely implausible}

best fit. The values

_plotted

were obtained

_{by fixing}

the

_positions

at

_points

on the

_hcp

lattice found to be the best fit to the

_{(100) peak positions,}

and the

_amplitude

to

_correspond

to the

_{largest pixel}

value encountered. The

_spread

in the

aspect

ratios for these

_peaks,

even within the one

sample,

was

_{significantly larger}

than the

computed

standard deviations.

Figure

4 shows the

_pixel

values

_along

a radial cut

_through

the

_peak

of

_figure

2 as a

Fig.

3. -

Histograms

formed

by àdding

normalised Gaussians for each best-fit

_major/minor

axis ratio with a standard deviation

_equal

to that of the measurement. The

_frequency

scale is

_arbitrary

but the

same for both curves on the one

_{plot. (a)}

First-order and second-order

peaks

for the three cadmium

Fig.

4. - Radial cross-section

plot

of the

(100)

spot of the DMPE diffraction

picture

whose contour is shown in

_figure

2,

represented

as

piecewise

constant

pixel

values,

together

with best-fit Gaussian and Lorentzian functions

_(solid

curve, and broken curve,

respectively).

(exponent

100)

and Lorentzian

_{(exponent 1.5).}

In both cases the

background

value was

assumed fixed at 32

_by

the

background

subtraction program, but the centre

_position,

amplitude,

radial FWHM and

_tangential

FWHM were allowed to _vary as

_part

of the fit. It can

be seen that the Gaussian overestimates the FWHM and underestimates the

_amplitude,

while the Lorentzian underestimates the FWHM and overestimates the

_amplitude.

This was the case in

_essentially

all the

_{peaks analysed.}

When the

_exponent

was also allowed to _vary, the best fit in all cases was found for intermediate values of the

_exponent.

In

_particular,

the one-dimensional Lorentzian

_{(exponent unity)}

was never a better fit.

Figure

5 shows the residual error for the fit to the Gaussian versus the residual error for the

fit to the

Lorentzian,

plotted

as the rms value

_(i.e.,

the square root of the sum of

squared

residuals divided

_by

the number of

_points).

The values were in all cases

_{significantly larger}

than the

_background

_noise,

_indicating

_{the presence of}

_systematic

error. While individual cases can be found where the Gaussian or the Lorentzian is a better

fit,

overall this test does not favour either the

_{polycrystalline}

or the NH model. However the width of the best-fit Lorentzian was in all cases much

_larger

than the film or

equipment

_resolution,

so _{that pure}

negative power-law

behaviour as

_{predicted by}

the

_paracrystal

model is

_completely

incompat-ible with the observations.

Table 1

_gives

the fit

_parameters

relevant to the

_{polycrystalline}

model for the third area of

Fig.

5. - Plot of the best-fit Gaussian error versus the best-fit Lorentzian error for all 36 first-order diffraction spot measured. The units are the those of the video

digitiser,

and

correspond

to

approximately

1 % of the

_{peak height.}

Table 1. - Peak width variation with

_diffraction

order and extraction

_{of physical}

parameters

on

However the values for this ratio appear to be

_{systematically higher, by}

an amount

_larger

than the

_probable

error. All values lie outside the ± 2 u error

_bars,

and one lies outside

3 o,.

The characteristic domain

_{size e}

was calculated from the best-fit FWHM

using

equation

(A.3).

To test the NH model in the third area

_{of comparison,}

a Gaussian is fitted to the diffraction

spot

profiles,

and table II

_gives

the

_resulting

best-fit

_parameters.

On this model the

peak

width should be

_proportional

to the

_scattering

vector

_Q

of the diffraction

spot,

so that in column 4 the width ratio divided

_by

the

_scattering

vector ratio should be

unity.

In fact of the entries in this

column,

two lie within the ± a error bars and all lie within ± 2 a.

The dislocation

_density

in column 5 of table II is deduced from

_{equation (B.12) using}

the measured values of radial FWHM. In

_comparison

with table

I,

it can be seen that the number

of dislocations

_required

to

_explain

the

_spot

_broadening

on the NH model is much smaller than

the number of domains

required

on the

polycrystalline

model.

Table II. - Peak width variation with

_diffraction

order and extraction

_{of physical}

parameters

on the Nelson and

_Halperin

model.

4. Discussion.

4.1 THEORETICAL. -

_Although

Nelson and

_Halperin

do not derive an

_expression

for the

scattering

function,

they

do derive the

_{asymptotic expression}

for a related function

Cq(R)

as R tends to

_{infinity [26].}

_Cq(R)

is the Fourier transform of the

_peak

of the Patterson function centred at the

_real-space

lattice

_point

_R,

evaluated at one of the six first-order

reciprocal

lattice

_points

_{q :}

This would seem to

_imply

a Lorentzian

_profile

for the diffraction

_spots,

as

_opposed

to the Gaussian law derived in

_appendix

B. The

_discrepancy

may be traced to the evaluation of the

This is

_justified

_when

_{1 YI}

is

_small,

i.e. for the inner

_peaks

of the Patterson function

corresponding

to

_separations

much smaller than the average distance between dislocations. When the

_{approximation}

is not

_made,

the correct

_asymptotic

form for

_C,(R)

is obtained. However the outer

_peaks

of the Patterson function are so broad that

_they

_{merge into} one

another,

and at the observed normalised dislocation densities of

_roughly

10-

_4,

the detailed

asymptotic

behaviour of the Patterson

_peak

widths makes little difference to the observable features of the

_scattering

function.

The NH

_theory

is not the first ansatz-free

_analysis

of diffraction line

_broadening

due to dislocations. The

_theory

for bulk media has been treated in several

_reports

_{[30, 45-48],}

but the results are

_distinctly

different from the

_present

2D case. In

_particular,

in

_{equation (B.12),}

the

expected

radial diffraction

_spot

FWHM varies with the normalised diameter B of the selected diffraction area, whereas in the three-dimensional case the

_spot

_broadening

is

_independent

of B. For an

_infinitely

wide film and incident

_beam,

the

_predicted

NH

_spot

diameter

_diverges.

Physically,

this is related to the fact that a

_{uniformly spaced}

linear row of identical dislocations with

_Burgers

vector

_{perpendicular}

to the row

_{(a symmetrical grain boundary) produces}

a

finite

_change

of orientation but a strain field which

_{decays rapidly}

to zero. As a _consequence,

there is no stress field

_neighbouring

the row which

_might

cause other dislocations to move and hence introduce an

_{exponential shielding}

_factor,

such as in found in the

_{Debye-Hückel}

treatment of an

_electrolyte.

On a more

_{fundamental level,}

the

_divergence

is related to Peierls

proof

[49]

that

_long

range order in two dimensions cannot

_persist

to infinite

_separations.

However,

in

practice

this

_{logarithmic divergence}

is

_scarcely

noticeable. The smallest feasible value of B is

_roughly

2 x

104

for a 1

_pm-diameter

selected diffraction _area, and the

largest

roughly

2 x

106,

_giving

a _range of variation

_{of spot}

size with diffraction

_aperture

of less than 22 %.

The line

_{shape predicted by}

the NH

_theory

has been described as

_Gaussian,

and this is to a

good approximation

true. However

_{strictly speaking,}

it is a Gaussian divided

_by

the square of

the distance to the

_origin.

This results in a shift of the

_spot

centre towards the

_origin,

and hence

_{larger d-spacings,}

with

_respect

to the

_position

for a defect-free lattice.

The NH model is

_{physically incompatible}

with the

_{polycrystalline}

_model,

because

_they

correspond

to two

_completely

different _ways of

_organising

the dislocations of the

_crystalline

lattice. However the NH model is

_compatible

with the

_{paracrystalline}

model,

which describes the line

_broadening

due to unbound lattice defects of a different

_type.

When both

_types

of defect are

_present,

the

_resulting

_spot

_profile

is the convolution of the two individual

_profiles.

4.2 EXPERIMENTAL. - The

_broadening

of the diffraction

_peaks

is in all cases much

higher

than any

possible

limit

_{imposed by}

the electron

_microscope.

The

paracrystal

model

_predicts

a

power law

peak profile,

with zero FWHM. Hence it is

_possible

to rule out the

_paracrystal

model,

at _{least in its pure}

_form,

and to _{say that the}

_crystal

lattice of the

_monolayer

must contain dislocations. This is the case in the other two models considered :

_they

are either

organised

in

_{specific configurations,}

as in the

_{polycrystalline}

_model,

or almost

randomly

distributed,

as in the NH model.

It can be seen from

_figures

2 and 3 that the contour lines are not

_circular,

as would be

expected

for the

_{polycrystalline}

model

_analysed.

The observed

_aspect

ratios for the well defined

_{(100) peaks}

show a

_{sharp clustering}

in the

_vicinity

of the value of

v3

_{predicted by}

the NH model. This was also the case in an

_{independent study}

of diffraction

_spot

intensities from

monolayers

of cadmium soaps of three different

_fatty

acids

_[44].

The

_{polycrystalline}

model could

_perhaps

be amended to include a statistical fluctuation of the domain orientation to restore _agreement with the

_experimental

_findings.

It is however difficult to see how such an ad

hoc modification could account for the observed

_clustering

of the

_aspect

ratios of the

₍₁₀₀₎

On the

_{polycrystalline}

_model,

the best fit characteristic domain

size e

is of the order of

5 nm, that is to _say 10 lattice constants. Now it is known that the

_density

of dislocations

along

a

_{grain boundary}

is related to the

_change

of orientation across the

boundary.

From the

diffraction

_patterns

it can be seen that the standard deviation of orientational fluctuation cannot exceed 0.05 radians. The difference in the orientation between two

_neighbouring

domains must on _average be less than twice this

value,

leading

to an

expected

number of

dislocations on domain boundaries of order

_unity.

However an initial

assumption

of the

model was that the interior of each domain was

_strain-free,

and efficient

shielding

of external

strains

_by

a

_{grain boundary requires}

_many dislocations. Hence the fit to the

_{polycrystalline}

model is

_internally

inconsistent.

Since it leads to a

contradiction

to assume that the

_boundary

of a

_grain

shields its interior

from external

strains,

the notion cannot hold for the

_{present system.}

On a

_suitably

modified

version of the

_{polycrystalline}

model, therefore,

the orientation within a

grain

is not

necessarily

constant. As

_already

discussed,

the observed

_long-range

orientational order is

incompatible

with the presence in the films of

large-angle grain

boundaries. It is well known that

_{low-angle grain}

boundaries can be resolved into a _{sequence of}

widely-spaced

dislocations

[50].

Hence the

_{polycrystalline}

model revised to take account of the

present

experimental

findings

starts to _{look very much like the NH model.}

A recent _{paper has}

_reported

thé direct electron

_{microscope image}

of a cadmium

eicosanoate

_{monolayer showing}

a

roughly-circular region

ouf -

103

molecules free of

dislocations or

_grain

boundaries

_{[51 ].}

This is consistent with the

_present

fits to the NH

_theory,

which indicate a defect

density

of fewer than 1 dislocation _per

104 molecules,

but at variance

with the

_present

fits to the

_{polycrystalline}

model,

where in each case the characteristic domain

size e2

contains fewer than

102

molecules.

The NH

theory

can

_provide

an

_explanation

for the observed

small,

but

significant,

departures

of the diffraction

_peaks

from

_hexagonal

_symmetry.

The

_divergence

of the

_spot

FWHM for infinite

_aperture

size is related to the fact that the strain field from a dislocation

decays

as

r-1,

with no cut-off. Dislocations

significantly

outside the diffraction

aperture

cause

the same strain at each

_{diffracting point,}

and so do not contribute to

_spot

_broadening,

but do

cause an overall shift in the

_{peak position.}

Since

_grain

boundaries in a

_{polycrystal effectively}

isolate each domain from any strain fields in

neighbouring regions,

the

polycrystalline

model

cannot

_explain

this

_phenomenon.

The deviations of

₍₁₀₀₎

_{spot aspect}

ratio from the NH

_prediction

were in the direction of

higher

ratios.

_High

_aspect

ratios were also found in an

_independent

electron diffraction

_study

of stearic acid

_monolayers

[10].

This is the effect

_expected

from a

density

of

disclinations,

which are observed in the films but which are not considered in the NH

_theory.

The

_density

of disclinations in

_monolayers

of 22-tricosenoic acid

_produced

in the same

_{laboratory using}

the

same

_equipment

as the

present

experiments

has been elsewhere

_{reported [52]}

to be of the order of 10 mm-

2.

A disclination of rotation 60° located 300 pm from the diffraction

aperture

of diameter 100 pm will result in a maximum relative rotation of the

_crystalline

lattice

by

20° from one side to the

_other,

_resulting

in an increase of the

_tangential

FWHM of an otherwise

sharp

(100) peak of approximately

0.3

_{cycles.nm - 1.}

Hence the excess

_{tangential broadening}

observed in the

_present

work can be

_{readily explained}

in this way. In a different

_laboratory,

one of the

_present

authors has measured much

_higher

disclination

_densities,

of the order of

1 000

mm-2

_[53],

so that much

_greater

deviations from the « ideal »

J3

ratio can also be

explained. Unfortunately,

methods of

_producing

_{aligned monolayers}

have not

_yet

been

perfected,

so that this source of

_uncertainty

cannot

_yet

be eliminated.

Although

some of the _spot

_profiles

observed in the

_present

work show smaller residual

form for the radial

_scattering

_profile

was also observed in two

_previous

studies

_{[9, 10],}

and is

incompatible

with the « _{pure » NH}

_theory

derived in

_appendix

B. _{However there appear} to be two

_plausible

_ways in which the

_theory

could be extended to

_explain

a

_proportion

of

non-Gaussian

_profiles.

Firstly,

the existence in LB films of lattice disturbances with zero

_Burgers

vector can

scarcely

be doubted. Reflection

_high-energy

electron diffraction

_(RHEED)

studies have

provided unambiguous

evidence for the coexistence of orthorhombic and triclinic

_aliphatic

subcell

_packings

with similar tilt in LB films

_{of pure long-chain}

acids

_[12].

It is also clear that

impurities

are almost

_always

_present

at the 1 %

_level,

and many authors have

proposed

that

g+

g- kinks

commonly

occur in the chains

_[54].

A further

_type

of disturbance with zero

_Burgers

vector which is

_indisputably

_present

is

provided by

the molecular

_shape,

which does not share the observed

_{macroscopic hexagonal}

symmetry.

In _{many smectic B}

_phases

it has been well established that the diameter of the smallest

_cylinder

which

_completely

contains the Van der Waals surface of one molecule is

considerably larger

than the measured lattice

_parameter,

_indicating

that the observed sixfold

symmetry

is the average over _{many small orthorhombic domains with orientation}

_{differing by}

120°

_[55].

This is also the

_present

case.

_Assuming

molecular dimensions of 0.154 nm for CC

bonds,

0.107 nm for CH

_bonds,

0.12 nm for the Van der Waals radius of H and 109° 28’ for all bond

_{angles [56],}

the minimum

_{enveloping cylinder}

diameter for an

_aliphatic

chain is

0.516 nm,

_{corresponding}

to a cross-sectional area _per

_hcp

chain of 0.23

nm2.

All such

_{paracrystalline}

lattice disturbances will lead to stress fields of the disclination

quadrupole

type.

As shown in

_appendix

_C,

it is

_possible

to combine the NH and

_paracrystal

theory

to form a

_theory

of lattices with

_independent

random distributions of both disclination

dipoles

and

_quadrupoles.

The

_resulting

_spot

_profile

is the convolution of the

_spot

_profile

due to

_{dipoles only,}

with that due to

_{quadrupoles only.}

Hence on this model the central

_shape

of

the

_peak,

_including

the

_FWHM,

is still determined

_by

the dislocation

_density.

When

paracrystalline

defects are

_present

in

significant

numbers,

the

_wings

of the

_profile

are

power-law

_(as

is the case in the Lorentz

_profile).

It is also

_possible

to

_explain

a non-Gaussian

_lineshape

in terms of a

_{non-equilibrium}

distribution of dislocations. It has

_previously

been

_reported

that the

_component

_{of optical loss}

due to

_{large-angle scattering}

in LB films of 22-tricosenoic acid has a

_square-law

variation with

the

_{in-plane optical anisotropy, suggesting}

that variations of local 2-dimensional lattice orientation are the

_major

cause of this

_{scattering [18].}

The

_present

_study

indicates that dislocations are a

_plausible

cause of these variations. There have been a number of

_reports

[57]

that the

_{large-angle scattering}

in LB films of several different materials is not

_uniform,

but is concentrated at certain

_points,

and that the

_density

of these

_{light-scattering points}

varies

systematically

with

_position

in the film

_[58]

and

_deposition

conditions

_[59].

Hence it is

plausible

that the dislocation

_density

in the films

_{displays macroscopic inhomogeneities.}

Clearly

it is

_possible

to

_modify

the NH

_theory

with a suitable statistical distribution of

dislocation densities to

_explain

the observed slower-than-Gaussian fall-off. Since

_spot

smearing

is also

_{accompanied by}

a

_displacement

of the

_spot

centre towards the

_origin,

this model also leads to an

_{asymmetric profile, falling}

off more

_slowly

on the side of the

undeflected

_beam,

_although

with the

_present

best-fit

_parameters

this

_asymmetry

is undetect-able.

5. Conclusions.

smectics. The diffraction

_pattern

to be

_expected

from a

_{Nelson-Halperin}

hexatic smectic has not

_previously

been derived from first

_principles.

The

_present

treatment of the

_paracrystal

is a

significant improvement

on

_previous

_attempts

in that the strain

_pattern

caused

by

the defects

is in mechanical

_equilibrium.

The

_predictions

of each of the three models have been

_compared

to

_experimental

_patterns

for LB films of two different

_aliphatic

chain

_{compounds, leading}

to the same conclusions for

both materials.

_Using

the

_aspect

ratio

_criterion,

the

_{polycrystalline}

model could be

_rejected.

The radial

profile

criterion

_provided

a definitive refutation of the

paracrystalline

model but

did not choose between the

_{polycrystalline}

and NH

_models,

each of which showed

_systematic

error.

_Although

the differences between the

_{polycrystalline}

and NH models for the width variation criterion were

_comparable

to the

_experimental

error, this test favoured the NH

model,

and moreover the best fit to the

_{polycrystalline}

model contained an internal contradiction. To sum _up, the best

_agreement

was achieved for the NH model of an

interacting

gas of

dislocations,

while the other two models had serious

_{shortcomings.}

Some

_{discrepancies}

in the fit to the NH model

remain,

and two _ways of

_improving

it have been

_proposed

for future

_{investigation.}

However,

independent

of the fine details of

interpretation, loosely

bound dislocations have been demonstrated to be a

_major

cause of

broadening

of the diffraction

_spots.

Acknowledgments.

This work was financed

_{jointly by}

the Bundes Ministerium für

_Forschung

und

_Technologie

within the Ultrathin

_Polymer

Film

_Project

_{No. 03 M 4008 C 3,}

_by

the Deutsche

Forschungsgemeinschaft

within the

_Special

Research Initiative SFB 262 on Nonmetallic

Glasses,

and

_by

the Materials Science Research Center of the

_University

of Mainz. The authors would like to thank Dr D. R. Nelson and Dr J. E.

_Riegler

for useful discussions.

Special

thanks are due to Professor H. Môhwald for

_having

initiated the

_project,

and for his continued

_{encourangement}

and

_support.

Appendix

A.

Dérivation of the

_{polycrystalline scattering}

function.

In the normal

_concept

of a

_{polycrystalline}

_material,

individual

_grains

are

_essentially

strain-free

perfect crystals,

with no correlation between the orientations of

_{neighbouring grains.}

When the selected diffraction area _{encompasses many}

_grains,

this

_{always gives}

rise to a

_ring

diffraction

pattern, contrary

to

_observation,

and

_contrary

to the

_optically

observed

_long-range

orientational order in LB films. Hence it is _necessary to consider an unusual case in

which,

through

some

_mechanism,

_{neighbouring grains}

have

_highly

correlated orientations.

However,

in order for the

_{polycrystalline}

idea to make sense it must be assumed that there is no

translational coherence from

_grain

to

_grain,

and that

_grain

boundaries are

_randomly

distributed.

The first

_step

in the derivation of the

_scattering

function is to calculate the Patterson

function,

or

_spatial

correlation function of the material refractive

_index,

from the assumed statistics of molecular

_position.

In the

_following,

_point

molecules with a delta-function refractive index

_profile

will be assumed. For

_{investigations}

of the lattice contribution to

scattering,

this is all that is

_required,

and in any case the

_scattering

function for real molecules

can be

readily

derived from that for

_point

molecules

_by

well-known methods. Within a

_grain,

the contribution to the autocorrelation function from a

_{given spacing}

is

_equal

to that for a

separates

the two

_points,

the contribution to the autocorrelation function is constant and

equal

to the lattice

_density.

As in the Poisson

distribution,

the

_probability

that no

_grain

boundary

separates

two

_points

varies

_{exponentially}

with the distance between them.

_Hence,

if M is the 2 x 2 matrix formed

_by

the unit cell vectors, then the Patterson function is

_{given by :}

Appendix

D defines the lattice comb function

_IIIM(D8)

and derives its Fourier transform

(D.9)

as well as that of e- r

(D.7).

The Fourier transform of the overall Patterson

function,

i.e.

the

_scattering

function,

is then

_readily

derived to be :

Hence

apart

from the

_{small-angle scattering peak,}

the diffraction

_spots

are all

identical,

with

circular cross-section and Lorentzian

_{profile (in}

the sense of power law

asymptote

- more

precisely,

Lorentzian to _{the power}

_1.5).

The characteristic domain

size e

is related to the FWHM

_{by :}

Appendix

B.

Derivation of the

_{Nelson-Halperin}

hexatic smectic

_scattering

function. The

_system

Hamiltonian is

_{given by [26, 60] :}

where

This is a bilinear form in the

_Burgers

vectors

_biP,

which have the dimensions

_{of length. i and j}

are lattice

_position

_labels,

while p, a =

1,

2 label the Cartesian

_components,

and

repeated

suffixes

_imply

summation

_using

the Einstein convention. a is the nearest

_{neighbour spacing}

of the

_hcp

lattice,

A is related to the Lamé elastic constants of the

_{mechanically isotropic layer}

by :

while 2

_{C /}

à,

_{the energy}

_required

to create a dislocation of smallest

_possible

Burgers

vector,

depends

in addition on the

_high-strain

nonlinear elastic behaviour of the lattice. On

physical

grounds

it is

_possible

to _say

_that,

since most lattice

_points

are not

dislocated,

then

C > kT,

while since the dislocations

_only

interact

_weakly,

then

Aa 2

kT.

In the

_{Debye-Hückel approximation}

the

_expectation

values of measurable

_parameters

are

calculated

_{by allowing}

the

_amplitudes

of the various

_Burgers

vector

_components

to _vary

continuously [26].

At thermal

_equilibrium

all

_{possible configurations}

of the b’s contribute to measurable

_{system parameters,}

with

_probability

or

_weighting

factor

_proportional

to

exp(-

H/kT).

This has the

_general

form of a multivariate Gaussian distribution of the b’s.