Scattering by a network of prolate, cross-section-polydispersed cylinders applicable to fibrillar thermoreversible gels

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Scattering by a network of prolate,

cross-section-polydispersed cylinders applicable to fibrillar thermoreversible gels

Jean-Michel Guenet

To cite this version:

Jean-Michel Guenet. Scattering by a network of prolate, cross-section-polydispersed cylinders applica- ble to fibrillar thermoreversible gels. Journal de Physique II, EDP Sciences, 1994, 4 (7), pp.1077-1082.

�10.1051/jp2:1994185�. �jpa-00248028�

Classification Physics Abstracts

61.10D 61.12B 82.70G

Short Communication

Scattering by

_a

network of prolate, cross-section-polydispersed cylinders applicable to fibrillar thermoreversible gels(*)

Jean~michel Guenet

Laboratoire d'Ultrasons et de Dynamique des Fluides

Complexes(**),

Universit4 Louis Pasteur, 4 rue Blaise Pascal, 67070 Strasbourg Cedex, France

(Received 23 March 1994, revised 17 May1994, accepted i June

1994)

Abstract. The intensity scattered by _a network of cross~section-polydispersed cylinders

in which the mesh size is far larger than the largest cross-sectional radius is calculated by considering _a weight distribution function of the type

w(r)

+~

r~~

_{with two cut-off radii,}

r~in

and _rmax. Two scattering _{ranges are} identified

(transitional

_range and

Farad-range).

The _case

I = I is particularly ^detailed _as it corresponds to existing experimental cases. It is shown that the different radii _{rmi~, rmax, rn}

(number-averaged),

_rw

(weight-averaged)

_can be determined

independently of the intensity absolute calibration.

1 Introduction.

The _numerous

investigations

carried _{out on} the

morphology of

thermoreuersible

gels

have shown

arrays of fiber-like _structures

[1.10].

^{While the mesh size of these networks is about I} ~m, the maximum fiber cross-section is but of _a few tens of nanometres.

Schematically speaking,

these structures may be

regarded

_as random assemblies of

prolate cylinders.

While in _{some cases} the fiber cross-sections _are defined in _a very narrow range of

values,

such _as ~-carrageenans for instance [10], ⁱⁿ many cases there is _a broad distribution of

cross-sections, _a

striking

and illustrative

example being poly-Irbenzyl glutamate gels

_[3].

The effect of

polydispersity

has

already

been worked _out for

chemically-formed

clusters characterized

by

_a fractal dimension D. Whereas

monodispersed

clusters should scatter _as

q~~,

it has been shown that

polydispersity

alters the exponent

D, yielding

_an apparent ^fractal dimension whose value

depends

_upon the chosen type of distribution

ill,

_12].

The aim of this short _paper is to

adopt

^the _same type ^of

approach

with cross-section-

polydispersed cylinders

_{so as} to work _out the

intensity

in the _range where it is affected

by

the cross-section

scattering

effects. This range is

usually

accessible

by

_means of

small-angle

neutron or

small-angle X-ray scattering techniques.

(*) ^This work has been supported by _a grant from the EEC

(Human

Capital and Mobility

Program)

enabling the creation of _a laboratories network entitled: "Polymer-solvent organization in relation to

c§din microstructure"

(**) CNRS URA 851.

1078 JOURNAL DE PHYSIQUE II N°7

2. Theoretical.

A

weight

distribution function

w(r)

for the

cylinder

cross-sections with two cut-off

radii, namely

rm~x and _rm;n is considered here. If in ^{is the network} mesh size, a simple

analytical expression

for the scattered

intensity

_can be derived under the

following

conditions:

gin

>

I(q

₌

47r/A

sin

6/2)

in $ _Tmax

The calculation _can then be achieved

by considering dilute, infinitely-long cylinders.

These conditions hold for _many thermoreversible

gels

in the

vicinity

of the critical

gelation

_concen-

tration.

2. I TRANSITIONAL q-RANGE. ^In this _q-range the transfer _{momentum range} is such that the

product

_{qr; can} take _any value

(hence

the _name transitional

q-range):

for low _r; values qr; ^< I while for

large

^values _qr; > I. No

simple

expansion of the

scattering

^function can

be carried out and one has to

integrate

_over the entire distribution to work out the scattered

intensity.

For infinite

cylinders

of cross-sectional radius _r; the absolute

intensity

reads [13]:

IA;(q)/C;

=

47r»L;Ji(qr;)/(q~ri) (1)

where C; is the concentration and _pL; is the _{mass per} unit

length (in g/molexnm)

of

cylinders

of cross-sectional radius _r;, and Ji ^{the Bessel function} of first kind and order I. If _{one assumes}

that all

cylinders

_are of the _same nature, then pL; is

proportional

to

r) _(pL;

#

ar)

=

7rpr),

where _{p is} the molecular

density

in

g/molexnm3).

For _a distribution of

cylinder

cross-section the intensity is written:

IA(q)/C

₌

4a7rq~3

_x

~j w;J)(qr;) (2)

;

where C is the

global polymer

concentration and _w; the

weight

^{fraction of} species

possessing

_a cross-sectional radius _r;.

Equation (2)

_can be written in the _case of a continuous distribution:

where W is:

~

W ₌

mm

^~~~

w(r)dr (4)

The

weight

distribution

w(r)

of interest here is:

w(r)

~

r-~ (5)

This

implies

that the total

length

of

cylinders

with cross-sectional radius _r in the

scattering volume, LT(r),

varies _as

r~(~+~J

_since

w(r)

r~

LT(r)r~.

The

integral

in relation

(3)

_can be evaluated

by

_parts

by subtracting integrals

from 0 to rm;n and from _{rm;n to cc} from the

integral

from 0 to _cc for which _an

analytical

solution exists

provided

0 _< ~ _< 3 [14]:

where T is the _gamma function.

For

intergrating

^from 0 to rm;n, the Bessel function _can be

expanded

for _{qr «} I

provided

that _arm;n < I.

Only

the first term needs then _to be taken into account.

Integration

^from

rmax to cc can be achieved

by expanding

the Bessel function for _{q~ »} I. The _terms of

higher

order of the

development

of the Bessel function _can be

dropped

_on the condition _qrm~x » I.

The

intensity finally

reads in _a

q~IA(q)

us. q

representation:

~

(~~~

^"

~j/ ^A(~)~~ _~~)~ ^~T$ilj (7)

~

max

As the transfer momentum range is such that _{arm;n <} I, the last term in

equation (7)

_can

now be

dropped provided

it is

negligible

with respect to

A(J)q'.

J = I is of

particular

interest _{as a}

simple

linear variation is

obtained,

_{a case} which pertains

to PVC

gels

_[15] and

poly-hexyl-isocyanate gels

[16]:

~~)~~~

=

27raq

^~~

x

Log~~ ~~'~j

for J

=

(8)

Tmax Tmin

Interestingly,

the

scattering

vector go for which

q~IA(q)

₌ 0 is related _{to rmax}

through

_a

simple

relation

independent

of the calibration

procedure

of the scattered

intensity:

2

~

0.637

~~~ ~

(g)

~~'~~

~rqo go

As will be _seen

below,

it is convenient to define the

following quantity

from the

slope

of relation

(8):

2.2 POROD RANGE. - If the inimum cross-sectional ^adius rm;n were _close to nought, then relation (7) would for the _scattering in the ^hole q-range.

On

account of _this

minimum

value,

there

exists

a q-range

where arm;n

» I. In

this domain,

Porod

domain, the

intensity

cillates round an

nction or qr

»

1,

n:

~~

~~

~

_8q~r2 ~ " ~ ~~~~

~l~~~ ~? /ll~ _II

~

Al

~~~~~~

~~~~

For J

= I relation

(12)

is

by neglecting (I/rm~x)3

with respect to

(I/rm;n)3:

~~)~~~

= 47rp x +

8q~r$;~Log~~'*~

~lfor

^{J = 1}

⁽¹³⁾

~n Tmin

where _rn is the

number-averaged

cross-section radius.

1080 JOURNAL DE PHYSIQUE II N°7

As _a

rule,

unless _very small radii _are dealt

with,

the second term is

negligible.

It is convenient to define the

following quantity

from the

plateau regime:

,~ =

q()j~~

= for ~ _"

(14)

Again,

for J

= I, _rn is written:

=

l~

x

Log~~ ~°~~~

for J

=

(15)

Tn Tmin Tmax Tmin

It follows from relations

(10), (14)

and

(15)

for J

= 1:

~'~~~

=

'°~'~~~

+ l for J =1

(16)

~min ~

As relation

(9)

allows _one to determine _~n~ax without

ambiguity,

_{rn~;n, rn} and _{p can} be also

straightforwardly

obtained. It should be further noted that the determination of _{rn~ax, rn~;n} and _rn

through

relations

(9), (15)

and

(16)

is

independent of

the absolute calibration.

Another parameter worth

calculating

is the

weight-averaged

cross-section

radius,

_rw.

rr~ax

Tw "

w m,n ^TW(~)d~

^{" (~max ~m>n) X}

L°g~~

(~max/~m>n) ^{f°~ >} " i

(17)

2.3 EFFECT _{OF CYLINDER} INTERSCATTERING. So far the theoretical expression for the

intensity has been worked out _on the basis of diluted

cylinders.

The condition

gin

> I implies that _one has _to consider all the

scattering objects

located within _a

sphere

of radius smaller than in. As its center is _not

fixed,

the

sphere

is liable to contain

"junctions"

_or

"crossings"

between fibers

(see Fig. I).

This will

give

_an additional

scattering (cross-terms)

which _needs to be evaluated.

Only

the terms between direct

neighbouring

fibers have _to be considered.

,""""'"

,' ,'

' 'I

i~

,

,'

',

Fig. 1. Schematic two-dimensional representation of _a fibrillar network. The differing line thick-

nesses schematize the different cross-sectional radii. The circles whose radius is q~~ illustrate the

range of distances explored. The dotted circle shows the _case where there is _a "junction" in the range of investigated distances. The condition qIn » 1 implies that only _one junction at _a time _can be in

this circle

(which

is _a sphere in the three-dimensional

space).

In the tral~sitional _{range one} is

dealing

with

infinitely-long cylinders.

Consider two

cylinders

whose

long-axes

_cross at point ^O. Be A and B two points _on

cylinders

1 and 2

respectively,

the

scattering amplitude

_can be written:

A(q)

= AD

exp(-iq DA)

₊ Ao

exp(-iq OB)

₌

Ai (q)

₊ A2

(q) (18)

Defining

the

angles

61 and 62 between the

scattering

vector and the

cylinder long

axis and

using

the

approach by

Fournet

ii?] eventually gives

for the cross-term:

~l (~)~2 (~)

j~/~ j~/~ ^sin(qLi

^cos

^{61)2Ji (qr}

^sin

^{61) sin(qL2}

^{cos 62 2Ji}

^(qr

^{sin 62)}

~

o o

qLi

_cos

61qr

sin 61

qL2

_cos

62qr

sin62 ^{~~ ~~ ~~~} ~~~~~~~~

(19)

If the

angle

between the _axes of the two-

cylinders

_were allowed _{to vary at} random, then the

integrals

in relation

(19)

could be calculated

separately. Yet,

_as the system is

frozen,

the

angle

between

cylinder

I and

cylinder

2 is fixed. Thanks _to this

condition,

it _can be shown that the cross-term vanishes for

infinitely-long cylinders

because the function

sin(qLcos6)/(qLcos6)

becomes

rapidly

_{zero except} for [18, 19]:

qcos 6 = 0

(20)

In other words, the

scattering

vector has _to be

perpendicular

to the

cylinder

axis otherwise the intensity

drops

_very

rapidly.

As the

probability

for two

neighbouring

^fibers to possess

simultaneously

the

right

orientation to fulfill relation

(20)

is very

low,

the cross-terms can be

ignored

in this _range.

The situation differs in the

Porod-range

in which the

infinitely-long cylinder

characteristic

plays

_no

longer

a role _as the system scatters as any

twc-density

medium. Cross-terms have to be taken into account as can be illustrated

by considering

the derivation made

by

Kirste and Porod [20]. ^For a medium with two main curvatures Cl and C2 these authors have derived the

traus>tioual PDrDd

q-range ....: q-range

q (nm~~)

Fig. 2. Example ^of experimental scattering pattern ⁱⁿ a

q~I(q)

us. q representation for PVC

/diethyl

oxalate gels (Cpvc ₌ 5 _x 10~~

g/cm~) [from

Ref. [15]]. This _{case can} be interpreted with 1

= 1.

From the intersect of the straight line obtained in the transitional q-range with the q-axis, _one derives

rmax " 10.6 _nm, which value agrees with electron microscopy findings (4]. ^{Use of} relations

(16)

and

(16) gives _{rm;n "} 2.2 _nm and _rn

= 4.4 _nm.

1082 JOURNAL DE PHYSIQUE II N°7

following

equation:

1(q)

_t

j

[1+

^<

~~~~

~

~/(~<

~~~ ~~~~ ~

(21)

in which the brackets denote the

averaging

_over all the

sample.

For _a

cylinder,

_one curvature is identical to _{zero so} that _one retrieves relations

(11). Yet,

when

cylinders

_cross one another both curvatures differ from _zero. This will result in

enhancing

the terms of

degree higher

than 4 _so

that,

in the

q~I(q)

_{us. q}

representation,

^the terminal

q~~

behaviour will be reached from above instead of

being

reached from below.

Figure

2 shows _an

illustrative, experimental

_case

(PVC gels

[15]) ^{for which J} = I

pertains.

3

Concluding

remarks.

Calculations

presented

in this _paper allow determination of the fiber cross-sectional dimensions

provided

that the

weight

distribution function of cross-sectional radii

obeys

a

simple

_power law.

Interestingly,

absolute calibration is not

required

^for

determining

the different cross-sectional radii. This theoretical

analysis applies

_quite

successfully

to the _case of PVC

gels [lS]

and

poly hexyl isocyanate gels

_[16] with J

= I. The

advantage

of the

scattering

method lies in its non-destructive characteristic _as it allows

investigation

of the

sample

in its

unperturbed,

three-dimensional _state unlike electron

microscopy which,

in routine

investigations, gives

a

distorted,

two-dimensional picture of the

gels. Despite

electron microscopy

shortcomings,

this

technique

is nevertheless _necessary to _ensure that the

gels

_{possess a} fiber-like structure which is _a

prerequisite

for

applying

the above equations.

Acknowledgements.

The author is

grateful

to M. Daoud for

helpful

discussions.

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