Self-association of disc-like molecules in hexadecane

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Self-association of disc-like molecules in hexadecane

Eric Y. Sheu, K.S. Liang, Long Y. Chiang

To cite this version:

Self-association of disc-like molecules

in

hexadecane

Eric Y.

_Sheu,

K. S.

_Liang

and

_Long

Y.

_Chiang

Exxon research &

_Engineering

Co., Annandale, NJ 08801, U. S . A.

. (Reçu

le 25 août _1988, révisé et

_accepté

le 23

_janvier

₁₉₈₉₎

Résumé. 2014 La self-association et les

_propriétés

_{thermodynamiques}

de molécules

_discotiques

dans l’hexadécane ont été étudiées à 22 °C par difusion de neutrons aux

_{petits angles.}

Les molécules étudiées sont le 2, 3, 6, 7, 10,

11-hexapentoxytriphénylène

(HET-5),

le 2, 3, 6, 7, 10,

11-hexanonoxytriphénylène

(HET-9)

et le 2, 3, 6, 7, 10,

11-hexa-undécyloxytriphénylène

(HET-11)

avec des concentrations

comprises

entre

10- 5

et

10-3

M. Tous ces

_systèmes

montrent une

tendance très

marquée

à la _{self-association dans la gamme de concentrations étudiée.}

_D’après

diverses

analyses

structurales, nous trouvons _{que les molécules forment des}

_{agrégats qui}

sont de forme

_{irrégulière}

à basse concentration et ont la forme de bâtonnets à

_plus

haute concentration. La distribution de taille dérivée du

_{premier principe}

_{d’ équilibre chimique}

a été utilisée dans

l’analyse

des

_agrégats

en forme de bâtonnets et on en a déduit les interactions intermoléculaires.

Abstract. 2014 Self-association and the related

_{thermodynamic properties}

of disc-like molecules in hexadecane were

_{investigated using}

small

_angle

neutron

_scattering

_technique

at T = 22 °C. The

disc-like molecules studied were _{2, 3, 6, 7,}

_{10, 11-hexapentoxytriphenylene}

_(HET-5),

_{2, 3, 6, 7,}

10,

11-hexanonoxytriphenylene (HET-9),

and _{2, 3, 6, 7,}

_{10-11-hexa-undecyloxytriphenylene}

(HET-11)

with concentration

_ranging

from

10-5

M to

10-3

M. The three _systems all show strong

propensity

of self-association at the concentration range studied. From various structural

analyses

we found that the molecules form

irregular

shape

of aggregates at low concentration and become rod-like as concentration increases. A size distribution derived from the first

principle

of chemical

equilibrium

was

_applied

to

_analyze

the rod-like _aggregate from which the intermolecular interactions was determined.

Classification

Physics Abstracts

60.61

_{- 61.12q}

- 61.12Ex

1. Introduction.

The

_aggregation

of molecules in solution is a common theme of interest in many areas of

science,

including biology,

colloid,

and

_polymer.

The most well studied

_examples

are

amphiphilic

molecules which exhibit rich varieties of structures such as

_{micelles, vesicles,}

and microemulsions

_[1-3].

In this paper, we

_report

the results of our studies on self-association

and the intermolecular interactions of disc-like molecules in oil.

_{Specifically,}

we have

performed

small

_angle

neutron

_scattering

_(SANS)

measurements to observe the self-association of disc-like molecules in

hexadecane,

and have

_analyzed

the results based on an

equilibrium

thermodynamic

argument.

The disc-like molecules have

special

architecture with

_rigid

_{aromatic groups} as the cores on

which flexible

_aliphatic

side chains are attached. In condense state, these materials exhibit

interesting

liquid crystalline phases, primarily

as _{the consequence of differences in}

_packing

requirements

for aromatic cores and

aliphatic

chains.

Typically,

the molecules stack into

infinite columns in various two-dimension close

_packed

lattices. Since the first

_report

of discotic

_mesophase

_[4],

extensive work has been advanced in

_synthesizing

molecules of different architectures and

_elucidating

their very rich

mesophase

behaviors

_[5].

So

far,

many studies on disc-like molecules are directed to the

_{understanding}

of the

_interplay

between the molecular architectures and the thermal

_ordering

behaviors of the

_{liquid crystalline}

mesophases

[6, 7].

The self-association of discotic molecules in solution was

_recently

observed via electronic

_absorption

and the

_optical

structure of the

_aggregates

were determined

_[8].

In this

_study,

we are interested

_mainly

in the interactions of discotic molecules in an

_aliphatic

medium. One

_{plausible implication}

of this

_study

is to understand the formation of carbonaceous

_mesophase,

which

_commonly

occurs in

_petroleum

and coal tar

_pitches

_{[9, 10].}

The carbonaceous

_mesophase

is the _{basic precursor for}

_{making important products}

such as

carbon fibers.

The paper is

organized

in the

_following

order. In section 2 a

_description

of the

_experiment

is

given.

In section 3 the size distribution function of discotic

_aggregates

is derived on the basis of chemical

_equilibrium

considerations. The SANS

_intensity

functions for different

_particle

sizes,

shapes,

and

_{polydispersities}

are

_given

in section 4. This is followed

_by

a detailed

descriptions

of the

_experimental

results and the data

_analysis

in section 5. In section 6 we

_give

the conclusion of this

_study.

_Appendix

A

_gives

a detail derivation of

_{intraparticle}

structure

factor for

_spheres

with size distribution

_according

to Schultz distribution function and

Appendix

B

_gives

that of

_{cylindrical particles}

with thermal

_equilibrium

distribution

_(to

be described in Sect.

₃₎

in size.

2.

Experiment.

The SANS

_experiment

was conducted on Beam Line H9-B at

_high

flux Brookhaven reactor at

the Brookhaven National

_Laboratory.

The

_wavelength,

A,

of the

_incoming

neutrons was

tuned to be 4.5

Â

with A À

_{/ A}

6 %

_using

a

_multilayer

Ge-Mn monochromator. The detector was an area helium-3 detector of 50 x 50 cm2

_containing

128 x 128

_pexels.

_By

_adjusting

the

sample

to detector

distance,

the

_{scattering intensity}

in the range of

scattering

vector

1 Q ( =

(4 -r /,k )

sin

_{( 0 /2 ), 0}

is the

_scattering

_angle)

from 0.018 to 0.17

Â-

can

be

_reliably

obtained.

The discotic

_samples

used in this

_study

were

_{synthesized by}

a

_{demethylation}

reaction

followed

_by

an

_alkylation

of a common intermediate of

_{2, 3, 6, 7, 10,}

11-hexamethoxytriphe-nylene

(HET),

as

_{previously reported}

_{[11, 12].}

The discotic molecules studied have

_alkyl

chains

_{Cn H2n +}

_{1 with n}

=

5, 9,

and 11. The solvent used for SANS

experiments

is

gold

label

deuterated hexadecane

_(Cambridge

Isotope

with deuterium atoms

_greater

than 99.8

_%).

In

mixing

HET-n with

hexadecane,

we found that

_samples

with concentrations in the range of

10- 4

to

10- 2

M were stable at room

_temperature

(see

Tab.

I).

However,

it is worth

_noting

that

by cooling

the

_sample

temperature

to below room

_temperature

thread-like

precipitates

could

be observed. This

_{precipitation}

_{process is} reversible. One can _{argue that the thread-like}

morphology

of the

_precipitates

is related to the columnar

_phase

of pure discotics. In

fact,

it will be shown later that this indeed is

_part

of the conclusions of this SANS

_study.

To

_perform

SANS

_{experiment, samples}

were loaded to circular

_quartz

cells of 1 mm

_path

length.

The scattered intensities were collected with a statistical error less than 5 %. The

measured

_intensity

was corrected for the

_scattering

from

empty

cell and the

_background

and

It will be shown later

_(see

Sect.

₅₎

that in order to differentiate different structural models in this

_study,

_{it is necessary} to use the absolute

_{scattering intensity}

in the data

analysis. Using

H20

standard,

the absolute

_intensity

can be obtained

_by

the

_following

relation

[13],

where

_subscripts

s stands for the discotic

solution,

w for water, and

_{(i, J)}

for

_pixel

number ;

ds/dn

is the differential cross section

(i.e.,

absolute

_scattering

intensity)

_{per unit volume of}

the

solution ; t

is the

_{path length}

of the

_sample

cell and T is the

_{corresponding}

transmission ;

I is the measured

_intensity

and M is the monitor counts.

3.

Equilibrium thermodynamic analysis

of disk

aggregation.

In the course of

_analyzing

SANS data from discotic

solutions,

we have

_incorporated

several

different kind of structural models for the

_aggregates.

As to be seen

_later,

the

_particularly

successful one is the one that assumes the

_aggregates

to be columnar

_shaped.

This model fits the SANS data for the

_high

_{concentration range}

_{( -}

3 x

10-3

to 1 x

10-2

_M,

see Tab. I and

Sect.

_5).

We therefore describe the columnar model in details first.

Let us consider the

_{thermodynamic driving}

forces for the disk-like molecules -to form

aggregates

in oils. In thermal

_equilibrium,

the

_system

contains monomers,

dimers,

and N-mer with certain number

_density

distribution.

_Considering

each N-mer as one

_phase

and let IL

and

UN be the chemical

potential

of a monomer and a N-mer

_respectively

in the solution.

Following

the Gibb’s

_equilibrium

_{thermodynamic}

_argument,

a monomer in any of the

_phases

presented

should have the same chemical

_potential,

i.e.,

By adapting

the common

_approach

used in micelle solutions

_{[14, 15],}

the chemical

_potential

for

_aggregates

of

_aggregation

number N can be written as

where

_{. N}

is the standard

_part

of chemical

_potential

and

_XN

is the mole fraction of N-mers. The second term of

_{(Eq. (3))}

_represents

the statistical

_entropy

of

_{mixing. Equation}

₍₃₎

is valid

only

for dilute

_systems

where

_{inter-aggregate}

interaction is

_negligible

and the

_mixing

is ideal.

By

substituting

(Eq.

(3))

into

_{(Eq. (2))}

one obtains

In

_(Eq.

_(4)),

_(X1)N

is the

_probability

of

_having

N molecules at the same

_spatial

location and the Boltzmann

factor,

exp[-

(N - N g’)Ik 1 B

T],

represents

the net _energy

_{gained by}

assembling

N free molecules in solution into an

_aggregate.

For discotic molecules the

_origin

of this energy

gain

comes from intermolecular interactions

(see

Sect.

5). Equation (4),

together

with the law of mass

conservation,

X

_{= 1 NXN}

(X

is the total monomer mole fraction in the

system),

are the two basic

_{thermodynamic}

_equations

for our self-association

analysis.

In order to formulate

_{(g " -}

_{N JL?)}

in

_{(Eq. (4))}

we

_adapt

the

_simple

laddér

model

[16].

In

this model the energy

gain

for

_adding

one molecule to a N-mer is assumed to be constant

_(i.e.,

dimer is then

_{( >§ - 2 IL?)}

= ô.

Similarly,

when N discotic molecules _are

stacked,

the _energy

gained

becomes

_{( > §l - NIL?) =}

_{(N - 1) 5. Accordingly, (Eq. (4))}

can be rewritten as

where

We will call the size distribution of the

aggregates

given

in

_{(Eq. (6))}

the thermal

_equilibrium

distribution from what follows.

_By

_{(Eqs. (6)}

and

_(7)),

the total molecular concentration can.

be

_expressed

in terms of 8 and

/3,

From

_{(Eq. (8))}

the

_weight

_average

_aggregation

number

_NW

_{and the number average}

aggregation

number

_NN

can be

_{conveniently expressed}

in terms of

_/3,

and

From these

_expressions,

one sees that in the ladder model

_i3

is the

_only

_parameter

needed to

characterize the thermal

_equilibrium

distribution. Once the value of

6 is determined,

6 can be obtained

_using

_{(Eq. (8))}

for a

_given

total concentration X. Since 8 is assumed to be a constant

in the ladder

model,

the size distribution

_XN

behaves like

_{f3 N}

_{(Eq. (6)).}

Therefore,

the

polydispersion

becomes

_{appreciable only}

if

_/3

is close to

_unity.

If the

_/3

value is

_greater

than

unity, XN

would then goes to infinite with N and the

_system

would be unstable

_(or

precipitation

would

_occur).

4. SANS

_intensity

functions.

For a

_monodisperse

_system

of

_particle

number

_density

_NP

and

_particle

volume

_VP,

the

scattering intensity

per unit volume of the

sample,

1 (Q),

can be written as

[17]

where

_pop

is the neutron

_scattering

_{length density}

of the

_particle

_{and p, is that of the solvent.} P

_(Q)

is the normalized

_{intra-particle}

structure factor

_(i.e.,

P

₍₀₎

=

1)

defined _{as square of the}

particle

form factor. The functional characteristics of P

_(Q)

are determined

_{mainly by particle}

shape

and size.

_S(Q)

is the

_{inter-particle}

structure

_factor,

which is

_{governed largely by}

the

interparticle

interactions. In our _case, such interactions are

_negligible

because the

_systems

studied are _{in very dilute} concentrations

_S(Q)

is therefore assumed to be

_unity.

The

_analytical

forms of P

_{(Q )}

are _{known for many}

_types

of

_particles

_{[18, 19].}

For the two relevant cases in

this

_{study, particles}

with

_spherical

and

_{cylindrical shapes,}

their P

_{(Q )’s}

are listed in

_Appendice

A and B for convenience of discusssions.

I(Q),

can be formulated by replacing NP,

V; and P(Q) in (Eq.

(11 )) with their weight

values

(Np),

( V)) ,

and

_{(P (Q))}

_{respectively,}

For a

_system

with

_given

_{particle shape,}

the

_{polydispersion}

can be

_represented

_by

the

distribution of the

_particle

volumes.

Thus,

(P(Q))

can be

_expressed

as

where

_{P (Q, V )}

is the

_{intraparticle}

structure factor of a

_particle

of volume V and

F (V )

is the

_particle

size distribution function. For

_{spherical particle}

_{P (Q, V )}

has

_explicit

form of

_{(Eq. (A.4))}

with V =

(4

-ff /3)

R3.

Similarly,

for

cylindrical

particles,

P

(Q, V) =

J

dlL

IF (IL,

L,

R) 12

with

_{F (1£,}

L,

R ) given

as

(Eq.

(B.7))

and V =

7rR 2L.

The

difficulty

in

analyzing

a

_polydisperse

_system

is often in the choice of a correct distribution function. In this

study,

we have

_investigated

two

_specific

models : one is the Schultz distribution for

_spherical

particles

and the other is the thermal

_equilibrium

distribution for

_cylindrical

_aggregates

which

we

_just

derived in the last section. In both cases, the

analytical

form of

_{(P (Q) >}

can be

derived in a

_{straightforward}

manner. The results are

_given

in

_Appendice

A and B

_(see

_Eqs.

(A.5)

and

_(B.6))

for both

_types

of

distributions,

respectively.

In the

_following

we shall derive

1 (Q)

for these two models :

A. SPHERICAL MODEL. - For

_{spherical particles}

with Schultz distribution in

_sizes,

the

intensity

contributed

_by

the

_particles

of radius

R,

1 (Q, R),

can be written as

where

_{P (Q, R) = [3}

_{jl (QR)/QR]2,}

_jl

is the

_spherical

Bessel function of the first kind.

_By

(Eq. (14))

the total

_intensity

_{1 (Q)}

can then be obtained

_by

_integrating

_{1 (Q, R)}

over R with

Np(R)

distributed

_according

to the Schultz distribution function. The

_explicit

form for

/(3)

is

_given

in

_Appendix

A

_(see

_Eq.

_(A.5)).

B. CYLINDRICAL MODEL. - For

_particles

formed

_by

_aggregation

of N molecules of volume

vrn, the

scattering intensity,

IN(Q),

can be written as,

_{neglecting S(Q),}

where

_CN

is the number

_density

of the molecules that form an

_aggregate

_containing

N

molecules,

PN (Q)

is the

_{intraparticle}

structure factor of such an

_aggregate,

and

_bm

is the total

scattering length

of a molecule.

Then,

the total

_intensity

I

_(Q)

can be written

_by

summation of

IN(Q),

By defining

1 (0)

to be

where

_Co

is the total number

_density

of the discotic molecules in the

_system.

_Using

_{(Eqs. (5)}

and

₍₆₎₎

for size distribution

_{CN/(N .}

_Co),

_(P(Q»

can be derived from

(Eq. (16))

to be

Equations

(16)-(18)

are the

_equations

to be used for

_analysis

of SANS data. The

_explicit

form

of

_{Pv (Q)}

in

_{(Eq. (18))}

for an

_aggregate

_containing

N stacked HET-n molecules

_(the

scattering

geometry

is a

_cylindrical

shell

(see

_Fig.

1))

is

_given

in

_Appendix

B

(Eq. (B.6)).

Fig.

la. -

HET-n molecular structure. The

_length

of the

aliphatic

side chains can be varied

_chemically.

Three different chain

_lengths

with n = 5, 9, and 11 _were

synthesized

for this

study.

Fig.

lb. - Structural model for stacked HET-N

aggregates of

_length

L. The concentric circle is the _top

view of the

cylindrical

aggregate which includes two

_{regions :}

the aromatic core

_region

and the

_aliphatic

chain

_layer.

The

_{corresponding scattering length profile}

show below indicates that the main contrast comes from the

_aliphatic

chain

layer

and solvent and the essential

_scattering

_geometry is a

_cylindrical

5.

_{Expérimental}

data and

_analysis.

The SANS measurements of the three sets of HET-n solutions

_{(Tab. I)}

all showed

_strong

small

_{angle scattering}

intensities.

_Typical

results are shown in

_figure

2 for HET-5 in the

Guinier

_plots

_{(i.e., ln 1 (Q) vs}

_{Q ).}

These

_plots

show no linear

_{region, indicating}

that either

the

_particles

are not

_{monodisperse spheres}

or the

_Q

_{range is}

_beyond

the Guinier

_region.

_Thus,

a more

sophisticate

structural

analysis

is needed. We also note

that,

by increasing

HET-5

concentration,

the

_scattering

_intensity

first increases

_(see

curves 1 and 2 of

_Fig.

2),

then

_drops

(curves 3

and

_4),

and

_finally

increases

_steadily

_again

_(curves

4,

5 and

_6).

This

behavior,

as

shown in the inset of

_figure

2 for the

_{scattering intensity}

at

_Q

= 0.03

 -1

for

HET-5,

_was

similarly

observed for HET-9 and HET-11 with the transitions

_{taking place}

in the solute

Fig.

2. - Guinier

plot

of HET-5 in hexadecane at various concentrations :

₍₁₎

1.3 x

10-4 M,

₍₂₎

6.5 x

10-4

M,

(3)

1.3 x

10-3

M,

(4)

4.3 x

10-3

M,

(5)

8.6 x

10-3

M, and

₍₆₎

1.3 x

10-2

M. At low concentration the

_{scattering intensity}

increases first and then

_drop

_{(upper panel).}

As concentration further increases the

intensity

scales as concentration

_{(lower panel).}

The intensities at

_Q

=

0.03 A -1

for

concentrations range of 1.3 x

10-3

M to 4.3 x

10-3

M for HET-5

_(9.26

x

10-4

M to

3.06 x

10-3

M for HET-9 and 8.4 x

10-4

M to 2.64 x

10-3

M for

_HET-11).

We thus define the concentration below this transition concentration to _{be low concentration range and that}

above to be

_high

_{concentration range. We attribute this} behavior to some

_type

of structural

transition to be discussed later.

We now discuss the results of our data

_analysis

in two

_{parts :}

the

_spherical

model and the columnar shell model.

A. SPHERICAL MODEL. - To

_investigate

the

_possible

structural model for the

_aggregates,

we

first

_analyzed

the SANS data

_assuming

the

_aggregates

to be

_spherical

with the size distribution

according

to a Schultz distribution. This distribution has been

_{widely applied}

in

_polymer

solutions

_{[20, 21].}

The

_analysis

was carried out

_by

_using

the formalism outlined in section 4.A and

_Appendix

A with three

_adjustable

_parameters,

_namely,

_average

_particle

radius

R,

the width

parameter

z of the

_{distribution,}

and an

_amplitude

factor. The

_amplitude

factor

was introduced because the

_{scattering length density}

of a

_spherical

_aggregate

formed

_by

discotic molecules can not be calculated a

_priori

as in the stacked

_cylinder

case

_(see

section

below).

Due to this additional

_parameter

the

_fittings

_may not be

_unique

and

_multiple

convergence may occur. In order to avoid

_multiple

_convergence we

_{systematically changed}

Rand

used

_only

z and the

amplitude

factor as free

_parameters

to fit our SANS data.

_Typical

results of the curve

_fitting

are shown in

_figures

3a and 3b for the case of HET-5 at low and

high

concentrations

_{respectively.}

The

parameters

(R

and

_z)

extracted from Schultz model

fittings

for all three sets of

_samples

are

_given

in table I.

For

_comparison,

the results of curve

_{fitting using}

a

_{monodisperse sphere}

model is also

shown in

_figure

3. One sees that the

_{monodisperse sphere}

model is not

_acceptable.

The

quality

of the

_fitting

is

_greatly

_improved

when the Schultz distribution is

_{incorporated.}

However,

the values of the width

parameter,

z, so obtained are

_relatively

low

(in

the range of

- 0.5 to

3,

see Tab.

_I).

The

_equivalent

width of the radius distribution is about 2.8 to 1.0 times of the mean radius. That is to _{say the}

_system

would be

_extremely

_polydisperse

Fig.

3a. -

Model

fitting

of HET-5 in hexadecane at 6.5 x

10- 4

M

using

various structural models. The

Schultz

_gives

the best fit. However, the intensities calculated

_{by using}

the extracted _parameters and the molecule-solvent contrast are

_{substantially higher}

₍

> 3

times)

than the measured

scattering

intensities.

Fig.

3b. - Same

fitting

as 3a for 8.6 x

10- 3

M. The Schultz

fitting

does not show as

_good

fit as it does for

the low concentration case shown in 3a.

_Again,

the calculated intensities are too much

_higher

than the measured absolute intensities. This _suggests that the

polydisperse spherical

model may not be

_applicable

for

high

concentration cases.

(p = 1/

Q

see

_Eq.

_(A.3))

based on this model. For such

_{high polydispersion}

the true

meaning

of

_spherical

_particles

can no

_longer

be assumed. It could

_equally

be

_possible

that the

system

contains

particles

with very

irregular shapes, particularly,

in the low concentration

cases. To our

_{experiences, spherical particles}

with Schultz distribution is a

_relatively

flexible

model and is able to accommodate the

_irregularity

of the

_{particle shape by decreasing}

the width

_parameter

_(or

increase the

_{polydispersity).}

One solution to this dilemma is to _compare

the absolute

_{scattering intensity.}

To do this we took the

_{particle-solvent}

contrast as that

between the discotic molecule

_(average

over core aromatic core and the

_{aliphatic chains)}

and the deuterated hexadecane.

_Using

the structural

_parameter

values extracted from

_spherical

Schultz model the absolute

_{scattering intensity}

were then calculated. The intensities thus calculated is much

_higher

₍

> 3

times)

at all

Q

than that obtained from measurements. We therefore believe that the

_{fitting using spherical}

_{Schultz model may}

_only

be

_{qualitatively}

meaningful.

We will come back to discuss this

_point

after

_showing

the results of the columnar shell model.

B. COLUMNAR SHELL MODEL. - Before we

_performed

the model

_fitting

we

_plotted

the

Porod

_plot

for the thin rod

_particle,

In

_{[ Q .}

I

_{(Q ) ] VS}

_{Q 2}

_(see

_Fig.

_4),

for the case of HET-5 to examine the

_large

Q

behavior. If the

_particles

are

_{monodisperse cylinder}

with

_large

_aspect

ratio the curve should show linear behavior at

_{large Q region.}

_However,

no linear behavior is

shown in

_figure

4 at _any

_Q

_region,

which

_readily

indicates that the

_particles

are not

monodisperse cylinders

of

_high

_aspect

ratio.

_Actually,

the

_nonlinearity

of

_figure

4 comes from the fact that most of the

_particles

are of low

_aspect

ratios. We will come back to discuss this

point

after structural

_{analysis using}

columnar shell model.

We now consider curve

_{fitting using}

columnar shell model for the

_aggregates

_{(see Fig. 1).}

Using

a CPK

precision

molecular model

[22],

the neutron

_{scattering length density,}

and thus

particle-solvent

contrast, of such an

_aggregate

can be calculated

(see

_Appendix

_B).

Fig.

4. - ln

[Q .1 (Q ) ] vs. Q 2

plot.

This

_plot

would show linear behavior at

_{large Q}

_region

if the

particles

are thin rods and are

_monodisperse

in size. It is obvious that no distinctive linear behavior

exhibits at

_{large Q}

for all the _spectra

_(1-6).

This is because most of the

_particles

are not thin-rod like and the

_particle

sizes are

_polydisperse

_(see

Sect.

_5).

computed

the

_{scattering intensity}

₇₍₆₎

_according

to

_{(Eqs. (15)}

and

_(16)).

The calculated

scattering intensity

was then fitted to the SANS data

_{using Q}

_(Eq.

_{(6)), d (Eq. (B.2)),}

and

Re

(see Fig. 1)

as

_adjustable

_parameters.

_Again,

we

_{systematically changed Re}

and used

_{only 8}

and d as

_adjustable

_parameters

in the

_fitting

so that

_multiple

_convergence can be avoided.

From the

_{extracted 6}

and d values we calculated the basic

_{thermodynamic}

_parameter

6

(the

formation

_energy)

_by

_{(Eq. (8))}

and the structural

parameter

L

(the

average

particle

length)

by

Fig.

5. - SANS data

fitting

of HET-5 in deuterated hexadecane of various concentrations :

_(D)

1.3 x

10- 2

M,

(A)

8.6 x

10- 3

M, and

₍₀₎

4.3 x

10- 3

M

_using

rod-like model with

particle

size distribution

according

to that derived in section 3. The

_qualities

of fits are reasonable but not excellent because

₍₁₎

the molecular

stackings

are

_{hardly perfect,}

₍₂₎

an _average

_{scattering length density,}

p a, was used for

_aliphatic

chain

_region,

and

₍₃₎

the aggregates are

_expected

to have some fluctuation in

shape.

Fig.

6. -

Same as

_figure

6 for HET-9 at

₍₀₎

9.26 x

10-3

M,

(.6)

6.1 x

10-3 M,

and

₍₀₎

Fig.

7. -

Same as

_figure

6 for HET-11 at

_(D)

5.28 x

10-3

M and

_(A)

2.64 x

10-3

M.

Based on these data

_fittings

we found that all the

_high

concentration cases in three

_systems

can be

_reasonably

fitted

_by

the columnar shell model at absolute

_intensity

scale.

_However,

the fits are

_quite

_{poor for low concentration}

_region,

which show abnormal

_{intensity-concentration}

dependence

(Fig. 2).

A

_{typical example}

is shown in the inset of

_figure

5.

C. THE PHYSICAL MODEL OF DISCOTIC AGGREGATION. -

_By

now, we have

_presented

the results of curve

_fitting

based on two different models in both low and

_high

concentration

regions.

The most

_promising

results are shown in the columnar shell model for the

_high

concentrations.

_Combining

these results with those from

_spherical

model

_fittings,

we

_might

argue that the molecules start to

_aggregate

at low concentration in a _very

_irregular

manner,

which results in the

_discrepancy

of absolute

_intensity

as we found in section

_V.A,

because the

molecular

_packing

are much loose than

_compact

_packing.

As concentration increases the discotic molecules start to stack

_regularly

and the

_aggregates

become rodlike. This amount to

saying

that there is a transition of

_particle

_geometry

such as that

_commonly

observed in

micellar solution

_(sphere

to rod

_transition)

_[23].

We consider a

_qualitative

_argument

that

_might

be

_applied

to

_explain

this transition. The association

_energies

of HET-n molecules are

mainly

derived from

dipole-dipole

and

quadrupole-quadrupole

interactions.

Thus,

the resultant structure of the

_aggregate

would

solely depend

upon the

competition

between these two forces. Our SANS data

_suggest

that the

_quadrupole

interactions

_predominate

at low concentration to lead the

_aggregate

structure to

_{irregular shapes}

via both vertical and horizontal associations. In this way, the rotational

entropy

and the chain-solvent

_affinity

dominate the free energy of the

system.

As concentration increases the

_{dipole-dipole}

attractions between aromatic cores become

predominent

and lead the discotic molecules to stack. The

_aggregates

thus form into

cylindrical shape.

Based on the

_{analysis using cylindrical}

shell model we found that all the values of the

parameters.

The

_/3

values extracted are very close to

_{unity indicating}

that the

_particles

are

polydisperse

in sizes. From this fact and that more small

_particles

_(low

_aspect

_ratios)

than the

large particles

in the solution at _any concentration

_(XN

is a monotonic

_decaying

function for a

given

/3

-

1,

see

_Eq.

(6))

one can

_{explain why figure}

4 fails to show linear behavior at

_{large Q}

region.

This is because Porod

_{plot requires large}

aspect

ratio for all

_particles

in the

_system.

The extracted dimer formation energy,

6,

was found to be

_nearly

constant

_{( - -}

13

_{KB T}

or

- 7.6

kcal)

within each discotic

system

which is consistent with the ladder model. The

_slight

changes

of 6 values as a function of concentrations

_{(from -}

13.01 to - 13.06

_KB

T for

HET-5,

- 12.61 to - 12.40

KB T

for HET-9 and - 12.26 to - 12.43

_{KB T}

for

_HET-11)

is attributed to

the

_changes

of intermolecular distance d’s

_(see

Tab.

_I).

One also notes that 8 value decreases with

_aliphatic

chain

_length

from HET-5 to

_HET-11,

which is

_expected

because the

_longer

the

aliphatic

chain the

_stronger

its

_affinity

to the solvent.

_Similarly,

the core-to-core

_{distance, d,}

increases with

_aliphatic

chain

_length

and the values

_{( ’"}

5.6 to 6.3

_Â)

are much

_greater

than that in pure discotic

liquid crystal phase

(3.6

Â)

[6, 7].

The

_NW

and

_NN

so obtained increase as concentration increases

_indicating

the

_growth

of the

_aggregates

_(see

Tab.

_I).

The cross sectional radius of the stacked

_{cylinder, Re, slightly}

decreases with discotic

concentration,

which may be attributed to the

_{increasing regularity}

of the molecular

_{stackings. Finally,}

we

approximately

estimate the dimer formation energy

by

calculating

the London forces

(neglecting

the

_quadrupole

forces and solvent

_effect)

between two HET-n molecules

(assuming

the molecule to be a

plate) [24].

Taking

Hamaker constant of benzene for HET-n molecule and

_inter-plate

distance to be 6

_Á,

the London force is at order of 15

_KB

T. With this

rough

estimation we believe that the formation energy extracted

(-

13

_KB

_T)

is

_quite

reasonable.

Recently

there was a luminescence

_experiment

carried out

_by

Markovitsi et al. on HET-6 in dichrolomethane solution at the _{concentration range similar} to our cases

_[29].

In this

experiment

no self-associations were observed unlike the results found in this

_study.

This is

because the solvents used are _{very different. Their}

_results,

_together

with our

_results,

_provides

a

_good

indication that the self-association is a _{energy driven process and its} occurrence

depends

on _{the energy}

_competition

between the intermolecular interactions and the molecule-solvent interactions.

6. Conclusions.

We have demonstrated that the discotic molecules

_{HET-5, HET-9,}

and HET-11 have

_strong

propensity

to self-associate in deuterated hexadecane. The

_aggregates

are

_irregular

in

_shape

at low concentration and become rod-like at

_higher

concentration. The columnar

_aggregates

are formed

_{by stackings}

of the disc-like molecules with inter-core distance - 6

Â.

The size distribution of these

_aggregates

is describable

_by

an

_equilibrium

_{thermodynamic}

model derived from the first

_principle.

The dimer formation

_{energies, taking}

into account the effect of molecule-solvent

_affinity,

are extracted to be about 13

_{KB T}

_(or

7.6

_kcal)

and

_{only slightly}

concentration

_dependent

which is consistent with the value estimated on the base of the

London force interaction.

7.

_{Acknowledgement.}

The authors wish to thank Drs. S. K. Sinha and C. R.

_Safinya

for useful discussions. One of

Appendix

A :

_Spherical

_particle

with the Schultz distribution.

Schultz distribution has been used

_extensively

in

_describing

the molecular

_weight

distribution of

_polymer

_systems

_{[20, 21].}

It is a two

_parameter

_{distribution,}

where

R

is the average radius of the

spherical particles,

z > - 1

[25]

is the width

parameter

characterizing

the

_degree

of

_{polydispersion}

of the

_particle

sizes,

and

_F(x)

_{is the gamma} function. Schultz distribution is skewed to the

_larger

sizes,

tending

to a Gaussian form at

_large

z value and

approaches

to a delta function as z

_{approaches infinity.}

Since

_j-th

moment of the Schultz distribution can be

_easily

calculated

_by

the

_{polydispersity}

index _p can be defined and derived to be

The

_{weighted intra-particle}

structure factor for

_{spherical particles}

with Schultz distribution in sizes can be

_expressed

as

_[25]

This

_integration

can be carried out

_analytically

as

_[26],

where

Appendix

B : Particle form factor of stacked disk-like molecules.

To

_compute

the

_{averaged particle}

form factor of an

_aggregate

formed

_{by stacking}

HET-n molecules in thermal

_equilibrium

we assume that this

aggregate

can be

_{approximated by}

an

effective

_{cylinder containing}

two

_regions

_(see

_Fig.

_1).

_Then,

the distribution of the

aggregate

region

of radius

_R,

and the

_aliphatic

chain

layer

of thickness t.

_Using

a CPK

_precision

molecular model

_[22]

the core radius and its volume can be calculated to be

4.4 A

and 184.9

_{Â3 respectively.}

The

_fully

stretched chain

_length

was calculated to be

8 A

for

_HET-5,

14 Â

for

HET-9,

and

16.6 À

for HET-11. The

_{corresponding}

volumes

_(volume

of six

_aliphatic

chains in a

molecule)

are

_respectively

933

Â3

1 680

Á3,

and 2 053

_{Á3. Figure}

la

_depicts

this structural model and

_figure

Ib shows the

_{corresponding}

_{scattering length density profile.}

It is clear from the

_{scattering density profile}

that the coherent

_scattering

is

_largely

contributed

_by

the

_aliphatic

chains and the

_scattering

_geometry

is

_essentially

a

_cylindrical

shell.

To calculate

_{PN (Q)}

of such a

cylindrical

shell stacked

by

N

molecules,

the average

scattering

length density

of the core,

Pc,

and that of the

_aliphatic

chain

_region,

p a,

have to be formulated. Since

_p,,,

_depends

on the core-to-core

_distance,

which has been

reported

to be

3.6 Â

in

_{liquid crystal phase}

_[6,

7,

27] (this

core-to-core _{distance may} not be retained in

_isotropic

solution

_phase)

we write

where L

bm

is the total

_{scattering length}

of the

core, d

is the core-to-core

_distance,

and

m

Re

is the core radius. L is the

_length

of the

_{cylinder relating}

to d and N

_according

to

or in the case of

_polydisperse

_systems,

where

3.04 Â

is the thickness of the aromatic core obtained from

_measuring

CPK

_precision

molecular model

_[22].

In

_aliphatic

chain

_region,

the

_{scattering length density}

is a function

of,

besides d,

the thickness of the

_aliphatic

chain

_{region, t,}

_(it

is not _necessary to be the stretched

length

of the

_aliphatic

chain for molecules in

_solution),

and r

_(Rc

- r -

Re,

see

_Fig.

_la).

Since the

_stacking

is

_expected

to be not

_perfect

and that there is

_{alkylchain flexibility,}

which was even observed in discotic columnar

_mesophases

[28],

we thus eliminated the r

_{dependence by}

taking

an

_{averaged scattering length density}

for this

_region,

where L

bm

is the total

_{scattering lengths}

of the six

_aliphatic

chains attached to the core and

m

is the shell volume. In

_equation

_(B.5)

Re

represents

the

_apparent

radius of the

_cylinder

_(see

Fig.

1b)

to be determined

_through

data

_fitting.

where 1£

= cos 8

( B

is the

scattering

angle)

and

F (ii,

L,

R )

is the

particle

form factor of _a

cylindrical

aggregate,

and

is the normalization

factor,

Apcf

=

Pc -

pf is

the

_scattering

contrast between core

_region

and

aliphatic

chain

_region

and

_dfifs

=

_{pre -}

P,

is

that between

_aliphatic

chain

_region

and solvent.

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