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From Guinier to fractalsRémi Jullien

To cite this version:Rémi Jullien. From Guinier to fractals. Journal de Physique I, EDP Sciences, 1992, 2 (6), pp.759-770.�10.1051/jp1:1992178�. �jpa-00246599�

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J. Pbys. I France 2 (1992) 759-770 JUNE 1992, PAGE 759

Classification

Physics Abstracts

6i.10-6i.40K-82.70D

nom Guinier to ftactals

R6mi Jullien

Bit- 510j Physique des Solides, Universit6 Paris-Sudj Centre d'orsay, 91405 Orsay, France

(Received 5 December 1991, accepted 13 January 1992)

Rdsumd En 1937, Andrd Guinier introduisit la m6thode de diffusion des Rayons X aux petitsangles qui est maintenant devenue

unoutil couramment utilis6 pour ddterminer la dimension

fractale d'agr6gats colloidaux. Dans cet article, on montre que la forme pr6cise de la fonction

de coupure (qui d6crit la fagon dont s'annulle, h grande distance, la fonction de corr41ation

interparticulaire) peut avoir une influence sur le comportement de l'intensit6 diflus6e par un

agr6gat fractal. Cet eflet, qui se manifeste dansun domaine de vecteurs d'onde situ6 entre

le r6gime de Guinier et le r6gime fractal, est misen

dvidence par des calculs analytiques et

numdriques.

Abstract. In 1937, Andr6 Guinier introduced the small angle X-ray scattering method

which isnow a

quitecommon

tool used to determine the fractal dimension of scale-invariant

colloidal aggregates. In this article, it is shown that the precise form of the cut-off function,which describes the large inter-particle distance decay of the pair correlation function in a fractal

aggregate, can have some influence on the behavior ofthe resulting low-angle scattering intensity.This effect, which takes place in a range of wavevector values located between the Guinier regimeand the fractal regime, is demonstrated by both analytical and numerical calculations.

1 Introduction.

In apioneering pre-war work (see Fig. 1), Andr4 Guinier, at the age of 26, has laid the

foundations of both experiments [I] and theory [2] of small angle X-ray scattering (SAXS).This powerful technique, which has been completed later

onby small angle neutron scattering

(SANS), is now used as a quite common tool [3-14] to extract the fractal dimension [15] of scale-

invariant fractal aggregates [16]. As already pointed out by Teixeira [17], the determination of

fractal dimension from SAXS (or SANS) experiments must be done witha great care. Here,

I show that the precise form of the cut-off which enters the interparticle correlation function

has an influence on the shape of the I(q) curvej I(q) being the scattering intensity and q the

modulus of the scattering wave-vector q, given by:

q =

)sin (i)

760 JOURNAL DE PHYSIQUE I N°6

137fj ACADIII>TIE DIIS SCIENCIIS,

R AYONS X. /,fi tlilfii.~.it>n i/L,.; iaj,t>nS .K >-(>as lit.< trds,faib/i~.r a>i@no.< appli-qu<'e d l'dtujiL' de Jines pailicu/t..~. et tic lid.ip<>nsions ccl/oida&S. Note de

II. Asnxk GuiNiEn, pidsen tie par hi. Diaries blau gum.

II est difficile d'dvaluer le diam4tre de la tacbe centrale qui semble

s'accroiti~e avec le temps de pose ; d'au ire part on De pent panel de la rgeurh mi-intensit6, puisqu'on ne connalt pas l'intensit4 maxima. Mars it e~tpossible de tirer du micropbotogramme des dam£es quantitatives prdcisesgrkce au

rdsultat tbdorique suivant :soient N atomes (au moldcules)

groupds enpetites particules toutes identiques de K atomes chacune; saris

faire aucune hypot1~4se sun la structure interne de la particule,on trouve

pour l'intensit6 diifoo4iaux angles tr4s foibles l'expression

-~~~£'Ji'

[ ~~h ~~~' ~ ~£'~' i

~

A est l'intensit4 dilfus4e par un atome, £l'angle de dilusion; et

K

R~= ) £r', r4tant la distance de chaque atome au centre de gravit4 de

i

la particule; R caractdrise l'dtendiie de la particule et joueun

rble

analogueau rayon de giration utilis6

enMdcanique. L'4tude de la

diffusion centrale donne done deux grandeurs:

d'abord la valeur de R,puts, si l'on

mesure l'intensit6 incidente et la masse du dilfuseur, le

nombre d'atomes moyen d'une particule;ces

deux r£sultans simultan£s

donnent des renseignementssur la forme des particules.

Fig. I. Reproduction of two parts (the title and the famous theoretical formulanow

knownas

the "Guinier law" of the second Guiiiier's articleon

SAXS[2] published in "Les Comptes-Rendus de

l'Acaddmie des Sciences, Sdance du 9 mai 1938". Ina

previous paper[I], the apparatus wasdescribed.

,I>here 9 is the scattering angle. Such effects take place inan

intermediate regime located

between the small-q Guinier law [2j 18] and the q~~ fractal law [16-17, 19], where D is the

fractal dimension of the scattered object. Thusj if the allowable range of (-values is not

sufficiently large, the estimation of the fractal dimension bya

simple power-law fit of In I(q)

as afunction of In q can be greatly affected.

N°6 FROM GUINIER TO FRACTALS 76i

2. Theory of wavescattering by

afractal object: the Guinier regime.

The theory of wave scattering bya

fractal object [16-17j 19] canbe viewed as a trivial extension

of Guinier's work on the SAXS theory [2, 18]. Ifone

considersan aggregate made with

alarge

number N of identical particles and if one canneglect multiple scattering effects, the scattering

intensity I(q)can be written as a double sum over the positions ri of the particle centers:

N N

I(q)=

lo ~ ~e"l (~'~~J) (2)

;=1 ;=1

In this formula lo is the scattering intensity foran

isolated single particle (form factor). HerejI

amconcerned by small-q and intermediate-q regimesj I.e. by q values smaller than the

inverse of the single particle diameter and, consequently, I will neglect the q-dependence of

lo- However, onehas to keep in mindj

asalready pointed out [17], that this q dependence

gives rise to strong deviations from the fractal regime at large q-values. In the following I(q)is always considered as being averaged over a

collection of statistically equivalent aggregates(thermodynamical ensemble).

Let me introduce in the usual way the pair correlation function C(r), such that NC(r)d~r

measures themean number of particles centered in an elementery volume d~r at a

distancer

froma

given particle. C(r) is normalized by:

fC(r)d~r =~~ ci1 (3)

Makinguse of C(r)j the double-sum

canbe transformed into

aspace-integral so that I(q)

appears to be proportional to the Fourier transform of C(r)

~)~=

N~ fC(r)e~~ ~d~r (4)

o

Assuming a spherical symmetry (this is true if either the aggregates areall spherically sym-

metric or ifone

deals with acollection of randomly oriented aggregates), C(r) depends only

on r = (r( and adouble angular integration can be done. I(q)j which now depends only

on

q =(q(, reads:

~~~~~=

4~N~ fC(r) ~~~ ~~

rdr (5)lo

o q

The low-q Guinier regime can be recovered by expanding (5) up to second order in q:

=N~(i )q~ /~ r~C(r)dr + ..) (6)

1now introduce the radius of gyration R, as being defined by:

N Noo

R~= m

L L(r; rj)~=

2~f r~C(r)dr (7)

;=1 j=1 °

Note that this definition is identical to the original definition of Guinier[2] (see Fig. I) where R~

is introduced as the mean of the squared distances from the center ofmass

(thiscan be shown

JOURNAL DE PHYSIQUE i T 2, N' 6. JUNE 1992 30


very quickly by putting the origin at the center ofmass in (7) and by expanding (r; r;)~).Making use of R, the expansion of I(q) reads:

j~~=

N~(1- ~+ ..) cy

N~e~~(8)

o3

which is, with different notations, makinguse of (I), and restricted to one aggregate (I.e.

K=

N in Guinier's formula), the formula depicted in figure 1.

3. Theory of wavescattering by a &actal object: the fractal reghne.

Starting from now, I will always refer to thesame

Guinier regime by using the set of followingdimensionless quantities:

f=

qRj ~ =

j; T(#)=

jj)~7(~)

=4~R~C(Rz) (9)

o

Thus, ~ and # measure lengths and wavevectors in units of the radius of gyration and its

inverse, respectively. With these notationsj the reduced scattering intensity is given by:

co

I(q)= =

sin(qz)~(z)zdz (10a)q

and the following two conditions, corresponding to formulas (3) and (5), must be satisfied:

fcoco7(z)z~dz

=7(z)z~dz

=2 (10b)

o

The last condition in (10b) ensures that the radius of gyration is equal to unity in the new

system of units.

In the case of a random fractal aggregate, onegenerally introduces the following form for

the correlation function [16-17, 19]:

7(z)=

z~~~~~~4(z) Ill)

The power-law dependence is imposed by the scale invariance properties of the aggregate (I

suppose here that the aggregate is built in the three dimensional spice)so

that D is the

fractal dimension[15-16]. #(z) is a cut-off function which must be necessarily introduced as

a consequence of the finite size (of order R) of the aggregate which breaks down its scale

invariance properties for lengths of order and larger than R. This function must tend to zero

morequickly than any power law for z » 1.

#(z)~-

const. for z « I; #(z)~-

0 for z » 1 (12)

Using this form, formula (10a) reads:

mI(f)= =

sin(qz)#(z)z~~~dz (13a)q

N°6 FROM GUINIER TO FRACTALS 763

with:~ ~#(z)z~~~d~

=I #(z)z~+~dz

=2 (13b)

To obtain the r~ fractal regime,one can make the change of variable y =

ix:

I(f)=

f~ /sin y #( ~)y~~~dy (14)

~

q

For f - c~o, one gets:I(f)

=Cf~~ (Isa)

with:~

C=

#(0) sin yy~~~dy (lsb)

One immediately gets aproblem for a fractal dimension larger than two since the integral in

(lsb) diverges. In such a case, one has to come back to the integral appearing in (14). Due to

the decreasing character of #(z) this integral converges, but it is absolutely not obvious that

its value is independent off or, at least, varying so slowly that this does not affect the r~

power law. Thus, it is not sure that the fractal law is still valid for D > 2 (This serious problemhas been already pointed out by Berry[20-21] and by Botet[19] in his thesis) and, even if it

is recovered (as it will be in all the following examples),one should expect some non-trivial

dependence of T(#) on the cut-off function.

4. The scattering intensity calculated withan

analytical cut-air function.

Before choosing an explicit form for the cut-OR function #(z),one has to keep in mind that this

function does not describe the cut-off of the density in real space. Some choicesare

forbidden,such as, for example,

asimple Yeaviside function (#(z)

=const. for z < a, #(z)

=0 for ~ > a),

which leads to possible negative values for I(#). Even foran object with

a sharp boundary,#(z) must exhibit some "width".

The most commonly used cut-off function [16-17, 19], which can be justified by comparisonwith critical phenomena [22], is a simple exponential:

4(z)=

Ae~~~ (16)

The two constants aand A can

be analytically determined by the two conditions (13b):

~

@'~ ID) ~~~~

where here r(z) denotes the gamma-function [23]. The whole function I(#)can

also beana-

lytically calculated (using mathematical tables [23] and is given by:

1(#)= (~) (l +

~)~) ~

sin ((D I)Atani) (18)D l T

~

T

~ ~

a

This is a well known and widely used formula [6, 8, 12, 17, 19]. In the limit #- c~o, one

obtains:I(#)

=

Cf~ (19a)


with:~D

,~D

"~2

~~~~

~~~~~

One observes that the fractal law is recoveredeven

for D > 2. However, itcan

be noticed

that for D=

3, I. e.for

a compact object, the above expansion is not valid andone gets q~~

instead of q~~ This result might be compared with the Porod-law [24] whichoccurs

when the

incident beam is scattered by the surface of an object rather than the volume. However here,

evenin the limit D

-3 the chosen cut-off remains smooth and it is not characteristic of

a

sharp boundary for the object.To see more

precisely how the q~~ power law is reached when D-

3, have plotted Ini

as a function of In#,as

given by (18), in figure 2a, for different D values. It canbe noticed

that for D=

2 the intensity isa pure Lorentzian since, in this case, formula (18) reduces to

T(#)=

~~_~. For D > 2, the

curveexhibits

a sigmoidal shape withan

inflexion point, after

the Guin~r~egime and before reaching the linear behavior with slope -D. This intermediate

region, which is character12ed bya

slope larger than D in absolute value, extends up to infinite

f when D-

3 and gives rise to the q~~ behavior when D=

3.

I have also madea

calculation witha cut-off function of the form:

#(z)=

Ae~~~~ (20)

In this case, a and A are given by:

D ~D-i Dqa = -; A

= ~(21)

~ ~(§)

and, using mathematical tables [23], I(f) is given by:

T(i)=

A(j)Mr(D 1)~~/ z(Di-~(i£1)) (22)

where Z denotes the imaginary part and Dp(z) is the "parabolic cylinder function" [23].Using the asymptotic properties of the function Dp(z) when

z - c~o, one finds, in the limit

f - ml

T(#)=

Cr~ (23a)

with:

C=

f~~~ji ~~ sin ()(D

))(23b)

r~_~~

Here again the expansion is not valid for D=

3. To determine the large f-behavior in this

case, one has to consider the next leading term of D(z) when z - c~o[23] to find:

T(I)=

2e~~l (24)

Thus the behavior recovered here in thecase D

=3 is completely different than with the

preceding cut-off. Note that, apart froma factor two, this is

asif the Guinier regime would be

prolongated up to infinite q-values.The corresponding numerical results are reported in figure 2b. Now, for D

=3 the behavior

isno more a power law but rather

an exponential decay and consequently the sigmoidal shapebetween the Guinier regime and the fractal regime is considerably

morepronounced than in

figure 2a.


o-o

-4.O

-+1,20

~Q-8.O 1.40

-1,60

1.80

-iz.o

-4

-16.o

2.90

2.98

~~~'~3.OO

-I.O O-O I-O Z-O 3.O 4.O 5.O 6.O

In ja)

o-o

-4.O

-+ 1.20

~Q-8.O

1.40'~

1.60

1.80

-iz.o z.oo

z.zo

-16.o z.60

3.OO2.80

z go

-20.O z_g8

-1.O O-O I-O Z-O 3.O 4.O s-O B-O

In j

bj

Fig. 2. Log-log plot of the reduced intensity fas a

function of the reduced wavevector #, for different

D values. Case a) corresponds to #(x)=

Ae~~~ (formula (18)) and caseb) corresponds to #(x)

=

Ae~~~~ (formula (22)).


5. The scattering intensity ofa

typical random fractal.

To test if the sigmoidal behavior found in both figures 2a and b is not an artefact of the specialanalytical forms chosen for the cut-off function, I have calculated the scattering

curve fora

"true" random fractal, built according to a simple iterative procedure known in the literature

as the fl-model[25]. One starts, at iteration p =0, from

acube of unit edge. At the first

iteration, the cube is enlarged bya

factor two in the three directions and is divided into 8

identical cubes of unit edge among which only n cubes are selected (I < n < 8) at random8j

(over all the, ~

possibilities) for the next iteration (the othersare

discarded). At theP.( P).

second iteration, the object is again enlarged by a factor two, each previously selected cube is

divided into 8 unit cubes, among which n cubesare

selected randomly etc.. At iteration p, one

gets afractal of size L

=2P, containing N

=nP unit cubes, whose fractal dimension is given

by:

~ ~~i ~~~~

In figure 3a and b, I show two typical examples, obtained after p =6 iterations (L

=64). Figure

3a corresponds to n =3 and figure 3b to n =

6 where the fractal dimensions are D ci 1.5849

and D ci 2.5849, respectively. Note that such fractals differ from realistic aggregates since the

subunits are not necessarily connected. However, it is known that only the mass repartition,and not the details of the connections, influences the scattering properties.

Once the fractal is built, one cancalculate I(f) directly by formula (2) which, after averaging

overall the orientations of the fractal in space, gives:

~~~~ /2 ~ $ ~~)~~~~~~

i J

where rip is the distance between I and j unit cubes. In practice, to reduce computing time, I

have first calculated g(k), the number of interparticle distances of length vi (k is an integerranging from I to km

=3 x (2P -1)~). Each distance is counted

once so that g(k) is normalized

by :

fg(k)=

~~(~ ~~ 127)

k=I

Then I have calculated I(f) using:

~ km sin~f

~~~~N

~N2 (~~~~ q@ ~~~~~

(f)

with:~

R~=

~ fkg(k) (28b)

k=i

I have reported the results obtained withn =

3, p =8, and with

n =6, p =

7 in figures4a and b, respectively (continuous lines). In case

a), for each chosen # value, T(f) has been

averagedover

100 independent simulations. In caseb) the average has been performed

over

two simulations only but I have checked that there was no big difference between these two


W-~~#~o~z-~lfi

$fl~fl~Q

jf.fldGOb$

t ~

~ i~jE~

iZ~

~ i~z(3

~ ~'~ Oit.I ~

did

~W%.I

] #

~.b °

$ 8#~tf

~ ~~q

~ GO ~$~f >

b0 ~'n.£

~ g~o$ ~~

~ W

~Qa 8~-'$#

# OU~~?C4

II

'a~Qt(~z~ j

~ CUD03#

'i )QI,I

II~~$QW'a W

3 ~ fl~$'Z~ «W ~ ~~ #.-*~~'~

~#q

$-~fl@

~~O'O~

~3 O

~ ~~g'O

~ fl~

~~i fl

&b~ tii~ # 8


simulations (the large number of particles, N=

279936, in this case already insuresa

goodaverage). The calculation done in

caseb) took 13 hours on an IBM 3090. Note that the large-q

departure from the fractal law occurs at lower # in case b) because the ratio between the s12e

of the object and the s12e of the subunits is smaller (the iteration number being smaller) than

in casea).

In both cases a) and b) the intensity curves have been compared with those, of same fractal

dimension, calculated with (18) (#(z)=

Ae~~~), with (22) (#(z)=

Ae~~~~) and with:

~~2~I(I)

"Ii + @)~~ (29)

This last formula, which was first introduced in the context of critical phenomena [26], has the

great advantage of simplicity.In both cases, one observes

amarked sigmoidal shape between the Guinier regime and

the fractal regime. This sigmoidal shape is even accompanied by oscillations which are more

visible in caseb) (where the fractal dimension is larger than 2). These oscillations

are certainlydue to the small thickness of the external surface of

ourfractals and are reminiscent of what

happens in the limit D=

3 ofa

sphere witha

sharp boundary [18] (in this extreme casethe

intensity vanishes, and its logarithm diverges, for certain q values). When comparing with the

analytical curves, it is interesting to notice thatnone

of them, except perhaps theone

with

#(z)=

Ae~~~~, does reproduceso strong an effect.

6 Conclusion.

In this article, I have shown that the precise form of the cut off function, which is characteristic

of the external surface of the fractal, has an influenceon

the shape of the scattering intensity

curve. Generally the Inl-versus-lnq curve presents asigmoidal shape with

aninflexion point

located between the Guinier regime and the fractal regime. Even if I have not donea systematic

study by varying the surface width, it seems that the sharper the boundary of the object, the

larger the absolute value of the slope at the inflexion point. This effectcan

bea source of errors

when trying to extract afractal dimension by fitting

anexperimental I(q) curve, especially

ifone

tries to include the Guinier regime within the fit. It must be known that the simpleanalytical formula (29) does not reproduce this inflexion. Even the

moresophisticated one

(22), which exhibits onlya

smooth inflexion for D > 2, might be inapropriate. The only hopeto make

areasonable estimation of the fractal dimension from a log-log plot of I(q) is to have

a

sufficiently large-q regime, well located inside the fractal regime, and to fit this part of the curve

bya

linear function, avoiding the inflexion region (as wellas

the large-q Bragg-like region). If

the available range of q-values is not sufficiently large, one canbe tempted to include part of

the inflexion region into the linear fit, but it must be known this, in this case, the resultingestimated fractal dimension would be systematically too high.


o-o

',

-z.o

'";

1-- '";

~ -4.O

'°;

-6.O

.°

-io.o

-I.O

a)~~ ~

o-o

i,;.

'k.-4.O

<.,'~=.

jM4 (.._

~ -8.0'"°

- .,

'",',",

-iz.o

,,

,

-16.O

-zo.o

-I.O O-O I-O Z-O 3.O 4-O 5.O B-O

~In j

Fig. 4. Log-log plot of the reduced intensity fas a

function of the reduced wavevector # numerically

calculated for the two fractals depicted in figure 4 (continuous curves). Case a) and b) corresponds to

fractal dimension and iteration number D=

1.585, p =8 and D

=2.585,p

=7, respectively. In case

(a) (resp. (b)) 7 has been averaged over 100 (resp. 2) fractals (built independently). In both cases

the intensity curves arecompared with those obtained with the analytical formulas (18) (dash-dotted

curves), (22) (dashed curves) and (29) (dotted curves), using the corresponding values for the fractal

dimension.


Acknowledgements.

would like to thank Robert Botet and Denise Tchoubar for interesting discussions.

References

[ii Guinier A., C-R- Acad. Sci. 204 (1937) 1115.

[2] Guinier A., C-R- Acad. Sci. 206 (1938) 1374.

[3] Schaefer D. W., Martin J. E., Wiltzius P. and Cannell D. S., Phys. Rev. Lent. 52 (1984) 2371.

[4] Axelos M., Tchoubar D., Bottero J. and Fiessinger F., J. Phys. France 46 (1985) 1587.

[5] Axelos M., Tchoubar D. and Jullien R., J. Phys. France 47 (1986) 1843.

[6] Freltoft T., Kjems J. K. and Sinha S. K., Phys. Rev. B 33 (1986) 269.

[7] Aubert C. and Cannel D. S., Phys. Rev. Lent. 56 (1986) 738.

[8] Schaefer D. W. and Keefer K. D., Phys. Rev. Lent. 56 (1986) 2199.

[9] Dimon P., Sinha S. K., Weitz D. A., Safinya C. R., Smith G., Varady W. A. and Lindsay H. M.,Phys. Rev. Lent. 57 (1986) 5y5.

[10] Chen S. H. and Teixeira J., Phys. Rev. Lent. 57 (1986) 2583.

[11] Cabane B., Dubois M. and Duplessix R., J. Phys. France 48 (1987) 2131.

[12] Vacher R., Woigner T., Pelous J. and Courtens E., Phys. Rev. B 37 (1988) 6500.

[13] Martin J. E. and Wilcoxon J. P., Phys. Rev. A 39 (1989) 252.

[14] Kallala M., Jullien R. and Cabane B., J. Phys. France, to be published.

[15] Mandelbrot B. B., The Fractal Geometry of Nature, W. H. Freeman Ed. (San Francisco, 1982).[16] Jullien R. and Botet R., Aggregation and fractal aggregates (World Scientific, Singapore, 1987).[17] Teixeira J., On Growth and Forms, Fractal and Nonfractal Patterns in Physics, H. E. Stanley

and N. Ostrowski Eds. (Martinus Nijhofl, Boston, 1986) p. 145.

[18] Guinier A. and Foumet G., Small-angle Scattering of X-rays (John Wiley and sons, New York,1955).

[19] Botet R., Thesis, Orsay (1985).[20] Berry M. V., J. Phys. A 12 (1979) 781.

[21] Berry M. V., and Percival I. C., Opt. A.cta 33 (1986) 577.

[22] Egelstafl P. A., An Introduction to the Liqdid State (Academic, London, 1967) p. 211.

[23] Gradshteyn I. S. and Ryzhik I. M., Table of Integrals, Series and Products (Academic, New York,1980) p. 490.

[24] Porod G., Kolloid Z. 124 (1951) 83.

[25] Benzi R., Paladin G., Parisi G. and Vulpiani A., J. Phys. A 17 (1984) 3521.

[26] Fisher M. E. and Burford R. J., Phys. Rev. 156 (1967) 583.

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