THE OPTICAL PHASE PORTRAIT OF FERROFLUID BITTER PATTERNS

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THE OPTICAL PHASE PORTRAIT OF FERROFLUID BITTER PATTERNS

U. Hartmann, H. Mende

To cite this version:

U. Hartmann, H. Mende. THE OPTICAL PHASE PORTRAIT OF FERROFLUID BITTER PATTERNS. Journal de Physique Colloques, 1985, 46 (C6), pp.C6-279-C6-282.

�10.1051/jphyscol:1985648�. �jpa-00224903�

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n°9, Tome ^6, septembre 1985 page Cé-279

THE OPTICAL PHASE PORTRAIT OF FERROFLUID BITTER PATTERNS U. Hartmann and H.H. Mende

Institut fiir Angewandte Physik dev Universitat Munstev, Corrensstrasse 2/4, D-4400 Munstep, F.R.G.

Résumé - On a obtenu une descritption théorique de l'aniso-

tropie optique induite par champ de dispersion d'échantillons de liquide ferrohydrodynamique de Bitter sous forme de tenseur diélectrique effectif localement défini. Cela permet d'inter- préter les résultats obtenus à la suite de recherches effectuées, à l'aide de technologies optiques sensibles aux phases, sur des structures fines se trouvant dans les parois de domaines.

Abstract - Theoretical expressions for the stray-field-induced optical anisotropy of ferrofluid Bitter patterns are derived in terms of a locally defined effective dielectric tensor. This per- mits the interpretation of results obtained from the investiga- tion of domain wall fine structures by means of phase-sensitive optical techniques.

I - INTRODUCTION

The observation of magnetic colloid patterns - the well-known Bitter technique - has been so far the most important single source of experi- mental information leading to a fundamental understanding of the do- main structure. It finds its highest development in the employment of

ferrofluids in conjunction with improved optical techniques, such as the observation of the stray-field-induced birefringence / 1 / and the phase -sensitive detection of colloid patterns by means of a polariza- tion interferometer 12/. A detailed analysis and interpretation of the ferrofluid Bitter patterns observed by such subtle optical techniques requires a largely quantitative relationship between the actual surface microfield distribution of a specimen and the stray-field-modulated dielectric properties of the colloid layer. However, a rigorous elec- tromagnetic diffraction formalism involving an accurate treatment of both, the ferrofluid thermomechanics and the ultimate optical image formation, should still be considered an idle hope, but the general nature of the dielectric stray field portrait can be deduced from the magneto-optical behaviour of the ferrofluid under controlled field con- ditions.

II - THE FORMATION OF BITTER PATTERNS

The formation of colloid patterns on the surface of a ferromagnetic specimen is due to the forces exerted on each ferrofluid particle by the inhomogeneous surface stray field distribution. According to /3/

the statistically averaged field-dependent magnetostatic energy <U> of a single particle is given by

<U>(p) = - kT p L(p) = - kT In p"

sinh(p) . (2.1) L(p) = coth(p) - p denotes the Langevin function with the argument

p = (XgmH/kT, where m is the magnetic dipole moment. It should be noted that the ever present particle size distribution and Neel relaxation

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985648

JOURNAL DE PHYSIQUE

effects, as they were discussed in /4,5,6/, are neglected in the deri- vation of (2.1). However, it is assumed that these approximations are not likely to change the general nature of the following results.

As far as the formation of ferrofluid Bitter patterns is considered as a thermodynamic equilibrium process the density variations within an ensemble of ferrofluid particles are governed by simple Boltzmann statistics:'

where @(p) is the field dependent volumetric particle fraction and

@O

@(p=O) the particle concentration at points with zero field.

~ h u s , (2.2) coincides with the classical Kittel relation / 7 / . I11 - CALCULATION OF THE DIELECTRIC STRAY FIELD PORTRAIZ

In addition to the density variations of the ferrofluid characterized by (2.2), the surface microfield distribution causes ordered arrange- ments of ferrofluid particles which lead to a locally induced optical anisotropy of the colloid layer /l/. The propagation of an electro- magnetic wave in the layer can be described in terms of an effective

dielectric tensor €(p), provided that the wave length X is large com- pared to the structural dimension of the particle arrangements / 8 / .

A Cartesian coordinate system (x,y,z) iq adopted with its z-axis in the direction of the local stray field H. A second Cartesian system (5,n,<) is assumed to coincide with the principal axes of an ellipsoidal ferrofluid particle which is subject to H. The coordinate transformation from (<,n,<) to (xryIz) is given by

cos8 cos@ sin@ sine cos@

cos8 sin@ cos@ sin6 sin@

-sine 0 cose

where 8,@ represent the polar angles of the <-direction yith respect to the (x,y,z)-system. The electric excess displacement D of the inner area of+the particle with respect to the carrier liquid is re- lated to the E-vector of the incident wave by

(D ,D ,D )

E O E (E E ,E

S n r ; E r n z; ^(3.2)

Using i,j

< , ~ , r ; , the dielectric tensor E

(Eij) is given by

zi

EC (Ep- - 1

Eij = Eidijl €Cl (€c

( E ~ - E ~ ) Nil , (3.3) where the complex numbers and

represent the dielectric constants of the carrier liquid and o f the fgrrof luid particle, respectively.

The Ni's denote the geometrical depolarization factors of the parti- cle. If the c-axis of every particle in an ensemble coincides with its easy axis of magnetization, the effective dielectric tensor €(p) of the colloid layer is locally defined with respect to the (x,y,z)-sys- tem by

where "1" represents the identical tensor. The local ferrofluid den- sity @(p) is determined by (2.2). The expression in angular brackets denotes a statistical average over all polar angles 9 and 4, where

U; + is the adjoint matrix to ug,@. After performing the averaging calculation the diagonal elements of €(p) can be expressed as

E~(P)=E (p)

+ @(p) r

2 (E -E ) p-'

Y i 5 r l r 5 ( 3 . 5 )

whereas all off-diagonal elements are equal to zero. Experimental in- vestigations of the magnetic and magneto-optical properties of ferro- fluids have provided evidence that it is appropriate to approximate the individual ferrofluid particles by prolate spheroids /5,6/. If the major and minor axes of a particle are denoted by a and b, respec- tively, the depolarization coefficients are determined by

Na + 2Nb

1, 2

Na = (l-e ) (2e3)-I ln[(l-e)(1-e-~]-2e, (3.6) with the eccentricity e

(1-b2/a2) Now, (3.5) simplifies to

(p) = E (p)

+

(p) [gb

(Ea-zb) P-' ~ ( p ) ] Y

Ez (p)

+ @(p) [ga - 2 P-' '(p)]

where Ca ^E and gb Tg=En. (3.7) characterizes the dielectric anisotropy arising within the colloid layer owing to the stray-field- induced orientation of individual particles. However, we suppose that

- due to the strong inhomogenities of the microfield distribution and the resulting high local particle concentrations - the interaction of the particles has a significant influence on the dielectric anisotropy of the layer. If a stray-field-induced aggregation of ferrofluid par- ticles into long rod-like chains is assumed, the resulting dielectric anisotropy can be calculated from (3.7) using the depolarization fac- tors Na

0 and Nb = 1/2 :

In the optical region the magnetic permeability tensor of a ferrofluid degenerates to p = 1. For i = x,y,z the indices of refraction ni and the extinction coefficients ki are related to the complex dielectric constants ~i by the Maxwell relation:

n i = R ~ ( E ~ ' / ~ ) , ki = 1rn(zi 1 /2) (3.9) The stray-field-induced optical anisotropy of a ferrofluid Bitter pat- tern is characterized by the birefringence An = nz-n and by the line- ar dichroism Ak = kz-kx.

Finally we should discuss the case of conventional colloid pattern analysis, i.e. the local dielectric anisotropy is neglected and only dielectric inhomogeneitiesresulting from the density variations of the ferrofluid are detected. For an isotropic accumulation of spherical particles (3.7) yields

where the only H-dependence is embodied in the local particle den- sity O (p) .

It should be noted that H represents always the magnitude of the vector sum of the local stray field component and an eventually ap- plied external magnetic bias field. Thus, the application of a homoge- neous bias field can provide useful modifications of the dielectric structure of the colloid layer, from which additional informations about the local stray field distribution can be derived. Referring to this, a striking example is given in /2/.

IV - OPTICAL ANALYSIS OF THE DIELECTRIC STRAY FIELD PORTRAIT

The employment of a polarization interferometer, as it was presented

in / 2 / , permits a phase-sensitive analysis of the ferrofluid Bitter

patterns. The stray-field-induced phase structure gradients are trans-

formed into contrast variations, whereas the ultimate image formation

C6-282 JOURNAL DE PHYSIQUE

depends on the dielectricinhomogeneitiesof the colloid layer as well as on its local optical anisotropy. Figure 'l shows an example for the resulting interference contrast micrograph of an iron whisker sample.

The topological formation of various Bloch walls is clearly revealed with a high resolving power. However, the ultimately observed image depends on the above discussed characteristic values of the local optical anisotropy as well as on the actual experimental arrangement, i.e. on the geometrical orientation of the walls with respect to the polarization directions of the incident light rays. This fact is es- pecially well demonstrated by the slightly tilted wall segment marked by the arrow and by the complete change in contrast at adjacent 90"

walls.

Fig. 1 - Typical high-resolution interference contrast micrograph of the ferrofluid Bitter pattern on an iron whisker sample at zero ap- plied field.

Acknowledgement

The authors wish to acknowledge the support of the "Deutsche For- schungsgemeinschaft" .

References

/l/ Hartmann, U. and Mende, H.H., J. Magn. Magn. Mat. 41 (1984) 244 /2/ Hartmann, U. and Mende, H.H., J. Phys. D: Appl. ~ h s . , to be

published

/3/ Hartmann, U. and Mende, H.H., 2. Phys., to be published

/4/ Hartmann, U. and Mende, H.H., phys. stat. sol. (a) 82 (1984) 481 /5/ Hartmann, U. and Mende, H.H., J. Magn. Magn. Mat. 45 (1984) 409 /6/ Hartmann, U. and Mende, H.H., Phil. Mag., to be pumished /7/ Kittel, C. Phys. Rev. 76 (1949) 1527

/8/ Born, M. and Wolf, E.: Principles of Optics (Pergamon, New York,

1980) 6th ed., P 705