X-RAY AND MAGNETIZATION STUDIES OF THE GEOMETRICAL PARAMETERS OF THE GRAINS OF A FERROFLUID

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X-RAY AND MAGNETIZATION STUDIES OF THE

GEOMETRICAL PARAMETERS OF THE GRAINS

OF A FERROFLUID

R. Anthore, C. Petipas, D. Chandesris, A. Martinet

To cite this version:

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X-RAY

AND MAGNETIZATION

STUDIES OF THE GEOMETRICAL PARAMETERS

OF THE

GRAINS

OF

A

FERROFLUID

R. ANTHORE and C. PETIPAS

UniversitC de Rouen, Laboratoire de Rayons X (*), 76130 Mont Saint-Aignan, France D. CHANDESRIS and A. MARTINET

Laboratoire de Physique des Solides (**), UniversitC Paris-Sud 91405 Orsay, France

R&um6. - On prksente une ktude par rnicroscopie Blectronique, par diffusion centrale des rayons X et par aimantation, de la distribution en taille, de la structure et des corrklations magnktiques entre les grains de colloides magnktiques. Pour les colloides B base d'eau (Fe304 dans l'eau), les grains se cornportent comme des sphkres dures disperskes dans un fluide porteur. Pour les colloides

B

base de toluene (grains de cobalt) on propose un modkle oh le grain est constituk d'un noyau de cobalt pur entourk d'une couche non magnktique. Dans les colloi'des de cobalt, des corrklations ont kt6 observkes qui mettent en kvidence une pkriodicitk dans la distance entre grains. Une forte anisotropie magnktique est observke pour les deux types de colloides.

Abstract.

-

The size distribution, structure and magnetic correlations between the grains of magnetic colloids are studied by electron microscopy, small angle scattering of X-rays and magnetic measurements. For water-base colloids (Fe304 in water) the grains behave as hard spheres dispersed in a carrier liquid. For toluene-base colloids (Co grains) a model of a grain constituted by a core of pure cobalt and a non magnetic sheath surrounding it is proposed. In toluene-base colloids, correlations have been obsenred which reveal a periodicity in the intergrain spacing. Strong magnetic anisotropy is reported for both colloids.

1. Introduction.

-

Ferrofluids are colloidal sus- pensions of monodomain ferromagnetic grains in thermal equilibrium in a non magnetic and insulat- ing fluid. They retain their liquid character in presence of high magnetic fields and resist firmly t o the building up of a large gradient of grain concentration in presence of a non homogeneous magnetic field [I].

We present a study of the size, the structure and correlations between the grains, by electron microscopy, small angle scattering of X-rays and magnetic measurements.

We work on two types of ferrofluids :

FLUIDS T.

-

They are magnetic colloids of cobalt in toluene. They have been prepared at Orsay by LiCbert and Strzelecki using the Hess and Parker's method [2] : they are produced by the thermal decomposition of cobalt octacarbonyl dis- solved in toluene and in presence of

a

terpolymer (methyl methacrylate (49.5 %), ethyl acrylate (49.5 %) and N vinyl pyrrolidone (1 %)). Statistical molecular mass is of the order of 14 000 and its radius of gyration as determined by small angle

by mechanical grinding. They are commercially available (I).

2. Electron microscopy. - A drop of the dilute fluid is dried on a sample grid covered with a carbon film.

Micrography of the A fluid reveals particles with sharply defined contours having a globular appea- rance.

The particles of fluid T are less contrasted, and present a quasi spherical form (Fig. 1).

scattering of neutrons is 40

A.

_FIG._{1 .}

_-

_{Electron rnicrography} _of _the _cobalt _grains _of A. - They are formed by dispersion in toluene-cobalt-magnetic colloids after evaporation _toluene. of the water as liquid matrix of magnetite grains obtained

(*) E.R.A. no 258.

(**) Laboratoire associ6 no 2.

( I ) Ferrofluidics corporation, 144 Middlesex Turnpike,

Burlington, Mass. 01803, USA.

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C2-204 R. ANTHORE, C. PETIPAS, D. CHANDESRIS AND A. MARTINET A quantitative analysis of these micrographies

has given the grain size distribution. The radius distribution for the spheres of cobalt shown on figure 2 obeys a normal law with a mean radius of 57.4 and a standard deviation a of 10.6

A.

FIG. 2. - Size distribution of the radii of the cobalt spheres derived from the electron micrographies.

For fluid A, the size distribution is well approxi- mated by a log-normal law but presents a large uncertainty due to the grain shape which is non perfectly spherical and their tendency to agglome- rate in clusters.

3. Small angle scattering of X-rays.

-

Small angle scattering of X-rays is a suitable technique for the study in situ of small particles from a few angstroms to a few hundred angstroms. The experimental set-up (Fig. 3) consists in a bend

FIG. 3.

-

Schematic representation of the set-up of the experiment of small angle scattering of X-rays ; S source of X-rays ; M monochromator ; F,, F2, Fp slits ; E samples ; C

linear counter.

crystal monochromator giving a beam of X-rays convergent, strictly monochromatic (Cu Kal radia- tion) satisfying to the conditions of the approximation of an infinite linear beam [ 3 ] . The scattered energy by the sample is measured in a proportional and linear counter which gives directly the scatte- red intensity versus the diffusion angle 2 8.

3.1 PRINCIPLE.

-

From the experimental inten- sity J ( s ) , one defines the normalized function jn(s) where s is the modulus of the diffusion vector :

v is the coherent energy scattered by one electron when it is irradiated by an X-ray beam of wave length A, 77 the sample thickness defined by the number of electrons per square centimeter, S I the

linear resolution of the detector given in units of the reciprocal lattice, Eo the intensity of the incident beam.

The electronic radius of gyration R , of the particles

141

is deduced from the slope of the linear part of the curve log (j,(s)) versus s2. If the scattering particles are spherical, homogeneous, one may define a Guinier's radius RG as

For monodisperse particles of radius R :

For a size distribution F ( R ) for the radii of the particles and a linear collimation of the beam, the Guinier's radius is given by

By extrapolating to zero the linear part of Log ( j , ( s ) ) versus s2, one gets the scattered intensity

jn(0) at the origin allowing to calculate one value

PG

of the volurnic fraction

p

occupied by the particles when the size distribution is known.

The value of the integrated intensity Qo is related to the mean square of the electronic density fluctuations by :

p electronic density in a given point of the fluid, jj

mean electronic density of the fluid.

The mean square of the electronic density fluctuations for our colloids is given by :

( P

- P ) 2

= P ( 1 - P ) ( p s - p 3 2

ps electronic density of the particle, PL electronic density of the liquid.

Let

Po

be the volurnic fraction of the scattering matter calculated from the preceding equation.

At large angles, the shape of the curve s3 jn(s) versus s reveals the presence or the absence of sharp edges in the scattering particles. If such a curve tends towards a non zero constant value, it exists a discontinuity in the electronic density between solvent and particles. From this value, one derives the specific surface S

/

V

of the particles and the Porod's radius R , which

is

proportional to the total volume V of the particles divided by the total surface area S :

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For a log-normal size distribution,

RG

and R, TABLE I

values derived from the small angle X-rays scatter- Volumic fraction of the magnetic material as determined by ing allow to calculate the two parameteks

R

and a various methods for T and A fluids. (For explanation of the

of that distribution. In effect, the nth moment with symbols see text). the log-normal law is given by [5] :

n

( R " ) = exp n

L O ~ R + - L O ~ ~ ~

2

Log R p = Log

R

+

2.5 Log2 o

The two last equations give m, and the frequency density of the particles given by :

F ( R ) = 1

6~

Log a

1

LO^

R -

L O ~ R

-2{

Log c

)3.

3.2 RESULTS. - The Log j,(s) curve versus s2

for the fluids T (Fig. 4) presents a good linearity which is characteristic of a small dispersion of the

0 5 10

FIG. 4. - Linear part of the curve Log(j,(s)) = f(sZ) for the T fluids.

particle size. The radius of gyration of the particles is 53

A,

their Guinier's radius 68

A

if we assume the grains spherical and homogeneous (Table I).

The same curve has been plotted for A fluid (Fig. 5). The linear part is shorter than previously, what means a broader size dispersion. The gyration and Guinier's radii are respectively 80

A

and 103

A. I

_s2 5

x l 0

0 ₅ ₁₀ &

FIG. 5. - Linear part of the curve Log(j,(s)) = f ( s 2 ) for the A fluids.

For the intensity j,(O) derived by extrapolation of the preceding curves (Figs. 4, 5), one may calculate the

PG

value for the volumic fraction occupied by the particles, value which has to be compared with that determined from absorption measurement of the incident beam,

pa,

(Table

I).

For T fluids,

Pa,,

agrees with the volumic fraction of the cobalt expected from the quantity of cobalt octacarbonyl chemically decomposed.

The agreement between

Pabs and

PQ,

is an indication that the grains scatter individually.

The difference encountered for the A fluid between the volumic fraction

PG and the other

values can be explained by the fact that PG requires a supplementary hypothesis on the sphericity of the particles while the others are not shape sensitive.

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C2-206 R. ANTHORE, C. PETIPAS, D. CHANDESRIS AND A. MARTINET

and the other values of

p

reveals that a fraction of cobalt does not contribute to the fluid magnetization as pure cobalt. This result has been encountered systematically for all T samples and the ratio

p ~ / p

= 0.57 varies only little.

The curve s3 jn(s) for fluid A (Fig. 6) is characteristic of particles without sharp edges and a broad

FIG. 6. - Curves s3 j.(s) = f(s) for the A fluids ; 0 experimental points ; .4 calculated values with a model of hard spheres with a log-normal size distribution in the approximation of an

infinite linear beam.

size distribution [6]. This curve tends towards a non zero limiting value indicating an electronic discontinuity between solvent and particle. The limiting value of s3 jn(s) leads to the Porod's radius Rp = 69.4

A.

The knowledge of RG and Rp determi- nes fully the repartition function, if one assumes a log-normal law. The characteristic parameters are

The assumption of a log-normal distribution for fluids A seems reasonable in respect to the mechanical grinding preparation of these fluids. The theoretical curve s3 jn(s) of figure 6 has been calculated for spherical particles of homogeneous electronic density with the preceding size distribution. The agreement between both curves is satis- factory. A model of hard spheres of magnetite coated by a surfactant and dispersed in a liquid matrix of water does apply for these fluids.

For T fluids, the experimental curves s3 jn(s) have a non standard form (Fig. 7). They tend towards a small limiting value. It still exists an electronic density discontinuity between particle and solvent whose value does not correspond to the expected discontinuity for pure cobalt and pure toluene. The no 1 curve of figure 7 giving s3 jn(s)

versus s corresponds to this pure case. Its limiting value is noticeably higher than the experimental value. This small value of s3 jn(s) may be interpreted by the presence of a sheath with a gradient of electronic density around a core of pure cobalt. Such a sheath of 15 A thick and a linear gradient varying from 2.4 e/A3 to 1.8 e/A3 gives the no 2

curve which approximates well the limiting value of

\.

.

E X P

FIG. 7. - Curves s3 j,(s) = f ( s ) for T fluids full line: experi-

mental results, dotted line no 1 : calculated curve for a model of hard spheres ; dotted line no 2 : calculated curve assuming the grain is constituted by a core of pure cobalt coated by a sheath

with a linear gradient of electronic density.

s3 jn(s). The oscillations which appear on the curves 1 and 2 are related to the first derivative discontinuity in the model choosen for the variation of the electronic density. The presence of such a sheath would also explain that a fraction of cobalt does not contribute to the magnetization, the volurnic fraction (p-PM) of non magnetic cobalt being in that sheath.

4. Grain correlations in presence of a magnetic field. - In the T fluids and in presence of a magnetic field parallel to the diffusion vector, the cvrve in(S)" - jn(') (Fig. 8) where j,(s)// is the

jn(s

1

T

exp -

t h e 0 - - - -

FIG. 8 . - Exverimental and theoretical curves

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The calculated curve of figure 8 which approximates the experimental one has been calculated with a model of 3 or 4 spheres aligned in chain and in equilibrium with spheres isolated or participating to non organized clusters. The fluctuations of the inter sphere distances in a given chain is of the order of 20 % and roughly 15 or 20 % of the spheres are involved in these chains.

For a field perpendicular to the diffusion vector, the scattered intensity does not present oscillations characteristic of a correlation length, but it is higher than the zero field one. This result may be interpreted by the presence of spheres which are correlated in the direction parallel to the field.

For A fluids, oscillations on the curve jn(s )/ - jn(s

1

have not been observed, due to the jn(s

1

broad size distribution for these grains : the correlations give very broadened oscillations which cannot be put in evidence.

5. Magnetic properties. - Figures 9A and B show the magnetization curves (which have been normalized for a 5 kG field) for :

-

the original fluids A and T,

- solidified samples in which the spatial distribution of the grains has been fixed in a magnetic field Hp (Hp = 0 for this figure) either, in the case of water-basis colloid, by substitution of the water by a polyvinyl alcohol [7] or, in the case of toluene-basis fluid, by polymerization of the matrix after transferring the grains in a mixture of styrene

o liquid

.

poiyrn.Hp=OG

.. Langevin (6 = 6 4 x )

I I I I I

o I 2 3 4 H(~G)>

FIG. 9. - Magnetization curve normalized for the value at

H = 5 kG, .versus H for the original fluid, a sample solidified in H , = 0 and a single Langevin function describing the monodis-

perse situation ; 9A : for T fluids ; 9B : for A fluids.

and divinyl benzene [8]. For comparison, a Lange- vin curve is shown. It describes the magnetization curve expected for a system of monodisperse and non interacting grains.

The fact that the magnetization curve of the sample solidified in zero field cannot be described by a single Langevin curve, reveals the grain size distribution [9]. This point is discussed in § 5.1.

The difference between the liquid and the solidified samples may be interpreted by magnetic correlations taking place between the grains. In the liquid, the magnetic correlations taking place between the grains. In the liquid, the magnetic field tends to align the grains while in the solid any displacement is impeded. The resulting anisotropy will be described in § 5.2.

5 . 1 GRAIN SIZE DISTRIBUTION. - We suppose a two parameters law F(R) for the distribution size, normal for T fluids or log-normal for A fluids as suggested by the electron-micrography measurements.

The magnetization curve is computed by super- posing various Langevin curves

where p is the magnetic moment of a grain, H the magnetic field, k the Boltzman constant, T the temperature, for different values of p, which is related to the size distribution by

Ms magnetization at saturation of the grain mate- rial, v grain volume, or, for spherical grains

R grain radius.

Each Langevin curve is weighted by a coefficient proportional to F(R).

The experimental curve may be described at

+

5

x

with the best fit as can be seen on figures 10A and 10B. One can thus derive the two parameters

a,

the mean radius, and a the standard deviation.

For the fluid A and a log-normal distribution

law (2), the results are :

-

R

= 49.7

A

a = 1.29

in good agreement with the X-rays and electron micrography determinations respectively 52.6

A

and 1.4.

As a matter of fact the accuracy is not sufficient to distinguish between a normal and a log-normal law and the magnetization curve can be described equally well by a normal size distribution with

-

R = 50.5

A

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C2-208 R. ANTHORE, C. PETIPAS, D. CHANDESRIS AND A. MARTINET

M

-

M (5 kG) 5.2 MAGNETIC ANISOTROPY.

-

The magnetiza-

1.0

-

tion is measured with a Foner vibrating sample

-0 -. o -O

,"lo magnetometer. In the fluid, this gives automatically

the magnetization parallel to the field (MI/). To

I

/

-

EXP. T' reach the parallel and perpendicular susceptibilities,

e calc normal d~stribution

qii

=33% we make use of the solidified sample technique.

o = 8% _{Typical curves are shown on figures 11A and}_B

i

for a field Hp of zero or 5 kG applied during the

solidification. The magnetic field modifies the A

.

I

isotropic spatial repartition of the grains which by magnetic interaction tends to make elongated clus-

[

ters. The magnetic anisotropy and thus the shape

anisotropy of the clusters, increases with the

o 1 2 3 ti

i

HI~GI applied field in the liquid.

RG. 10. - Fit of a magnetization curve with a superposition of Langevin functions taking into account the size distribution ;

10A : for T fluids with a Log-normal law ; 10B : for A fluids with a normal law.

M

For the fluid T, a normal law gives :

M ( 5 kGI

1 0 -

These values are very similar to those found by electron microscopy taking into account the presence of a sheath as indicated before from the X-rays measurements

(P 3.2)

: core radius = 57.4

A

- 15 = 42.4

A

M(H)

m!

-D -@ ,D/~--B 1 8 ' A02

2 = 0.25 for electron microscopy and X-rays 0.24 for magnetic measurements.

r:

-

Two remarks can be made about the fact that the radius determined magnetically is slightly slower than the other :

i) the preceding analyse assumes a spherical shape for the grains, what seems reasonable from the micrography, and assumes also that the magnetization at saturation Ms of the grain is the same as that of bulk Co ; 0 5.- 1 0Hp:5000G HIlHp 0.5 b Hp = 5000G H L H p .Hp = OG

FIG. 11. - Normalized magnetization curve of solidified samples for the direction parallel or perpendicular to the field applied during the solidification. For comparison, the magnetization curve for an isotropic sample (solidified in Hp = 0 G) is

shown ; 11A: for T fluids ; 11B : for A fluids.

.- '7

-

Exp

0 calc Log -normal distrlbutlon ,R = ~ 9 7 A

i=

-

/"

'0. 129 as

/

The spatial repartition of the grain is trapped by the matrix solidification. As the magnetization is sensitive to this spatial repartition, the fact that M,,,,(H = H,) = M ( H = Hp) solidified samples in H D

is a good experimental evidence that the repartition is not perturbed by the solidification technique [8], [lo].

The free energy of the system may be written as :

d M

+

aMZ(l - 3 cosZ

4 )

- 514 o Hp=50COG H//Hp a H : 5000G H 1 Hp P -

.

H ~ = O G

I

B

ii) the grain size is just lower than the critical

- MH cos ( 8 - 4 ) size for superparamagnetic behaviour and the

largest grains can have a different magnetization n number of grains, 8,

4

angle between the curve in the liquid and in the solid matrix. principal axis of the clusters (direction of Hp) and

>

(8)

H or M respectively, a is a numerical coefficient which is introduced by analogy with the case of linear chains as calculated by Jacobs and Bean [l 11 : for a chain of N spheres, the interaction term writes :

with

d distance between the two adjacent spheres. The linear case has been applied to ferrofluids [12]. Here we present an extension to the three dimensional case where the lateral interaction between grains are taken into account pheno- menologically through the parameter a .

Minimizing E versus M and

4,

one gets ~ ( 0 , a , H).

For low and high fields, we can make use of the approximations of the Langevin function (respecti-

U

vely X(u) -- - ₃ and X(u) = 1

-L)

_U and derive explicitely : For M / / low field M = H 1 - 4 a x o x T M s high field M = Ms - - p H + 4 e M s For ML1 low field M = H 1 + 2 a x o kT M s high field M = Ms - - p H - 2 a M s

xo

is the initial susceptibility of a sample solidified in zero field.

For each sample solidified in a field H,, a is determined by comparison of the experimental magnetization curve with a set of curves calculated for various values of a. On figure 12, the corres- ponding values of a are plotted versus the magnetization M of the sample for a field equal to the field applied during the solidification. Phenomenologi- cally and although the error bar is large, it appears

FIG. 12. - Values of the cu parameter describing the magnetic anisotropy for A fluids versus the value of the normalized magnetization M for a field H = H, equal to the field applied

during the solidification of the sample.

that the anisotropy is roughly proportional to the magnetization of the sample.

6. Conclusion. - The determination of the geometrical parameters has been obtained for different magnetic colloids :

-

For fluid A, the parameters of the log-normal distribution have been determined by electron microscopy, small angle scattering of X-rays and magnetization measurements. The results are consistent with a description of the colloid by hard spheres of magnetite dispersed in a carrier liquid, the water.

-

For fluid T, the X-rays results can be interpreted with a model where the grain of cobalt is formed by a core of pure cobalt surrounded by a non magnetic sheath of

-

15 A thick. The normal size distribution of the grains is found by the different methods with approximately the same width and mean radius, although a slightly smaller radius is found by the magnetic measurements.

Correlations have been observed for T fluid by X-ray scattering, i.e. it exists a periodical arrange- ment of the graihs over 3 or 4 sljheres in a direction parallel to the field.

The anisotropy in the magnetic susceptibility is reported. It is related to the spatial distribution of the grains which tend to make acicular clusters in presence of a magnetic field.

References

[I] KAISER, R. & MISKOLCZY, G., I.E.E.E. Trans. Magn. 6 [7] CHANDESRIS, D., MARTINET, A. & STRZELECKI, L., to be

(1970) 694. published.

[21 P.H. gL. PARKER' P. H' Jr.' J . Appl' Sci.

[S] LIEART, I,., MARTINET, A. & STRZELECKI, L., J. Colloid

(1966) 1915.

[3] LUZZATI, V., Acta Crystallogr. 13 (1960) 939. Interface Sci. 41 (1972) 391.

[4] GUINIER, A. & FOURNET, G., Small angle scattering o f [91 JACOBS, 1. S . & BEAN, C. P., J. ~ p p l . phys. 27 (1956) 1448. X-rays (New York, Wiley) 1955. [lo] CHANDESRIS, D., Thesis Orsay, to be published. [5] NEILSON, G. F., J. Appl. Crystallogr. 6 (1973) 386. [ l l ] JACOBS, I. S. & BEAN, C. P., Phys. Rev. 100 (1955) 1060. [6] TCHOUBAR, D. & MERING, J., J. Appl. Crystallogr. 2 (1%9) [12] BURGER, J. P. & MARTINET, A., Proc. Int. Conf. Magn.