Theory of magnetic suspensions in liquid crystals

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Theory of magnetic suspensions in liquid crystals

F. Brochard, P.-G. de Gennes

To cite this version:

F. Brochard, P.-G. de Gennes. Theory of magnetic suspensions in liquid crystals. Journal de Physique,

1970, 31 (7), pp.691-708. �10.1051/jphys:01970003107069100�. �jpa-00206970�

691

THEORY OF MAGNETIC SUSPENSIONS IN LIQUID ^CRYSTALS

F. BROCHARD and P. G. de GENNES

Physique ^{des Solides} (*), ^{Faculté des Sciences,} 91-Orsay (Reçu ^{le 20 mars} 1970, révisé le I S avril 1970)

Résumé.

On discute certaines propriétés ^de particules petites ^et allongées,

suspension ^dans

un

cristal liquide, à basse concentration (fraction

^volume f ~ 10-3). ^{On montre} que dans la

plupart ^des cas, la direction des particules

peut pratiquement pas s’écarter de l’axe nématique

local n(r). ^{Si l’on} parvenait ^à préparer ^{de telles} suspensions

^des grains magnétiques,

pourrait réaliser des suspensions

ferronématiques

ferrocholestériques» ^dans lesquelles, localement, ^{tous les} grains ^sont aimantés dans

même direction (parallèle ^à n) ^{et aussi des} matériaux

compensés»

aimantation spontanée. ^{De telles} suspensions ^{devraient montrer}

des distorsions et des changements ^de phase remarquables

présence de faibles champs ^exté-

rieurs H. On attend aussi des phénomènes ^de ségrégation : ^les grains ^{tendent à}

déplacer ^vers

les régions ^du spécimen ^{où n est} parallèle à H. Des effets analogues,

champ électrique, pour- raient

principe être obtenus

des suspensions colloidales de macromolécules polaires ^en bâtonnets, par exemple

des hélices de polypeptides.

Abstract.

The present paper discusses

properties ^of small, elongated particles suspended

in

nematic

cholesteric phase, with low concentrations (typical ^{volume fraction} occupied by ^the particle f ~ 10-3). ^We find that in most

the particle axis should be locked and parallel ^to

the local nematic axis n(r). If such colloidal suspensions ^{could be made} successfully ^with magnetic grains, ^{it should be} possible ^to prepare

ferronematics »

ferrocholesterics _», where locally

all particles

magnetized ⁱⁿ

^direction (colinear ^to n), ^{and also} « compensated » ^materials

with

spontaneous magnetization. These three types ^of samples should all show remarkable distorsions and phase changes in low external fields H. Similar effects

expected under electric fields with colloidal suspensions ^of polar rod molecules such

helical polypeptides.

31, 1970,

1. Introduction.

To observe magnetic ^effects

in liquid crystals, ^we usually require ^rather large fields H (- ¹⁰⁴ Oe) because the anisotropic part ^of the diamagnetic susceptibility xa is small ( N 10-’) ^[1].

We could try and increase _X. by introducing ^parama- gnetic ^{ions with an} anisotropic g factor. For strong spin ^orbit coupling, _{gll - g,} may be of the order of 0.3.

Assuming complete alignment ^{of the} magnetic solute, the anisotropic part ^{of the} paramagnetic suscepti- bility ^is given by

The number n of paramagnetic ^ions per CM3 ^cannot

be _very large, ^{if we want to} maintain the mesomorphic properties ^of the substrate. Typically ^we could reach

n

102°, giving XII - ^{xi -} 10-7 . Thus the _parama-

gnetic susceptibility ^{is not} efhcient. We _may increase the coupling by going ^to larger magnetic objects,

such as ferromagnetic grains. ^{It is not} easy ^to make

a colloid with such grains. But, ^{if this} turns out to be feasible in a liquid crystal matrix, ^the resulting

( *) Laboratoire associé

C. N. R. S.

(1) This effect

pointed ^{out to}

by ^{J. P.} Burger.

materials should have remarkable properties, ^which

we discuss in the present paper. How ^are the grains aligned ^{in the nematic ? We can think of two contri-} butions, ^one magnetic ^{and one} mechanical in origin.

ll) ^EFFECT OF THE GRAIN MAGNETIC FIELD.

Even for a spherical grain, ^the distribution of the field around the grain ^is anisotropic (Fig. 1). Therefore, there will be a preferential direction for the magnetic

moment of the grain ^{in a nematic} phase (1). ^{But this}

effect is weak for small grains. Because of the rapid

decrease of the field in 1/r3, the nematic configuration

FIG. 1.

Effect of the grain magnetic ^field

^the spatial configuration of the nematic molecules.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01970003107069100

is only perturbed ^{in the} neighbourhood ^{of the} grain.

The free energy, in presence of the magnetic ^field

of the grain, ^{is to first} order

where no(r) ^{is a unit vector} describing ^the unperturbed configuration [2]. ^F depends upon the angle ^between

the nematic axis no and the magnetic ^{moment u.}

The coupling ^term Fnp,/I - Fnp, 1., (the difference in

0 0

energy between the two nematic arrangements ^shown

1 u2

on Figure ^{g la and} lb) ^{is of} order 1 ₂ ay ^xa _V ^{,where Vis the}

grain ^volume. Typically X. - 10-6, ^{V = 10-16} cm3,

2

IL = 10 13 uem and _Xa JU v ^{10-4 eV « kB T.}

b) MECHANICAL _COUPLING, due to the shape ^{of the} grains. ^{We shall see} in section II that for an elongated grain ^with length L and diameter d(Lld - 10), ^the distortion _{energy of} a nematic liquid crystal ^is usually

minimal if the long ^{axis of the} grain coincides with the optical ^{axis. To} put the grain ^{in a} perpendicular

direction requires ^an energy of the order of KL, where Kis an average of the Frank elastic constants [2].

KL is of order of UL, _a ^{where U is a typical y mole-}

cular interaction and a is ^a molecular diameter.

Lla - 1 000, ^thus Emee» kB ^T. Therefore the coupling ^is mainly ^of mechanical origin ^{and is} very strong ⁱⁿ the case of elongated particles.

In section II of the present paper, ^we analyse ^first the long range distortions induced in a nematic matrix by ^a single particle. ^Then we move on to the

case of a dilute suspension, ^{and find} the minimum concentration cm;n above which the average nematic orientation follows the orientation of the grains. ^This

is roughly Cmin N 1/LD2 ^{where D is the} sample ^dimen-

sion. The corresponding filling ^factor f nin

^{Cmin V}

is thus of order [d/D]2 where d is the grain ^{diameter :}

thus fm;n is very small and in most experimental ^situa- tions f» fmin, i. ^{e. the molecular} orientation and the

grain orientation are along ^{the same} direction. A dif- ferent and more difficult problem ^{occurs at} higher

concentrations when the particles ^are magnetic ;

there is in fact ^a critical size above which different

grains ^{will tend to cluster because of the} magnetic

interaction. Let us ascertain what is the condition for weak clustering.

Consider rodlike grains (L/d - 10). They ^{will not}

make chains if

where E is the decrease in the demagnetizing ^field

energy when two grains ^stick together. To estimate AE let us replace the individual grains by ellipsoids ^of axial ratio L/d, ^{and the} two-grain system by ^an ellip-

soid of axial ratio 2 Lld. ^This gives :

where M, ^{is the} magnetization inside the grains, ^and

N a demagnetization coefficient [3], (N (10) ^{= 0. 24}

and N (20)

0. 08). ^{Because of the} exponential depen- dence of the left hand side in (I.1) ^{there are in fact}

two sharply ^defined regimes :

(1) ^Small grains : ^the inequality (1.1) ^{is satisfied}

for all feasible values of f (i. ^e. up ^to f = 10-2).

(2) Large grains : ^the inequality (1.1) ^{cannot be}

satisfied at any physical ^value of f.

Typically, ^with M,

^{103 and} L/d

10 the thresh- old between (1) ^and (2) ^{occurs at L = 2 000 A. In}

the present paper ^we restrict ^our attention to lengths

L below threshold.

In section III we discuss the _response of such _magne- tic suspensions ^{to an external} magnetic ^{field H ;}

there are two types ^{of effects :}

a) Distortions of the nematic (or cholesteric) pat-

tern. These distortions should ^occur in very low fields :

the _average magnetization ^{of the} suspension, ^{instead of} being Xa H (- ^{10-4 to} 10-3) ^{is now M}

M,/= 0.1

to 1. _{gauss ;} the gain ⁱⁿ coupling is thus of order 103.

b) ^Grain segregation : ^{we have seen that the} grains

have their moment locked along the local molecular axis n(r). ^Thus they ^{tend to move to} regions ^where n(r) ^is parallel ^to the field. These effects should give

rise to « depletion layers » ^of typical ^{thickness 1 to}

10 microns, ^{which could be observed} optically ^on

thin samples.

II. Floating objects ^{in a nematic} phase. -

II. 1 LONG RANGE DISTORTIONS INDUCED BY A SINGLE OBJECT.

We consider ^a solid grain, ^of arbitrary shape, floating ^{in a nematic} liquid (2). ^{The surface of}

the grain may be bare, ^{or it} may be coated by a suitable detergent increasing ^the grain solubility. ^{We assume}

that the nematic molecules near the grain ^{surface have}

a well defined orientation : specific examples ^will be discussed in section II.2. For the moment, ^we shall not investigate ^the complicated distortions of the nematic alignment which result in the vicinity ^{of the} grain, ^but only ^the long range part ^{of these distor-} tions : this turns out to be relatively simple, ^{and also}

to contain all the physical information which is useful in practice. We restrict our attention ^to static

properties (which ^are relevant for free energy calcula- tions) : ^the grain ^under consideration is fixed in

position (we put ^the origin ^of coordinates at the center of gravity ^{of the} grain) ^and in orientation.

As is usual with nematic systems, ^{we find a consi-}

(2) ^{To avoid} sedimentation,

require V(pg -pn)gD kB T where _p9 and pn

densities of the nematic and of the grain,

g

981

s-2 and D is the sample ^dimension.

derable simplification ^{if the} three elastic constants of Frank [2] ^{are taken to be} equal

in which case the distortion free energy of the nematic takes the simple ^form

where n is the «director, ^{i. e., the unit vector} labelling

the preferred ^axis.

II .1. a Definition of ^a « distortion amplitude ^».

Far from the grain (r > L) ^{n becomes} parallel ^{to the}

nematic axis no (We ^{take the}

^axis along no). ^More precisely ^we may put n(r)

no ⁺ ôn(r) ^where

is small. Eq. (II 1) ^{leads to a} free energy density

and to the equilibrium equations

It is convenient to put

where B12 Wx

’BJ2Wy

^{0. Since Wz is arbitrary we}

may also impose Bj2wz

^{0. Then the most general form}

of ^co at large r ^is

where a and P are respectively ^{a vector and a} dyadic, independent ^{of r. We shall} mainly ^{focus our attention}

on the leading ^term alr. ^{We note first} that, ^{in a} nematic, the states n and (- n) ^are identical. For this reason w and (x must be even functions of the unperturbed

director no.

We further restrict our attention to grains having

an axis of cylindrical symmetry (labelled by ^{a unit}

vector u). ^The grain shape may be ^a cigar (Fig. 2a),

a pancake (Fig. 2b), ^a tapered cylinder (Fig. 2c) ^{or a}

helical rod (e. ^{g. a} rigid polypeptide) ^{with a definite} sign of rotation (Fig. 2d). ^{The vector CI} depends ^on

both u and no. We shall prove first that a is an even

function of u. The argument ^{is indicated on} figure ^3,

where ^we keep only ^{the term oc} 1 /r ⁱⁿ eq. (11. 3) :

w is then an even function of r. Consider a point (1)

FIG. 2.

Various type of uniaxial grains.

FIG. 3.

Comparison ^{of the} distortions corresponding ^to

grain and the reversed grain.

in the (u, no) plane ^{and its} symmetric (2) ^{with respect}

to the origin. w.(I) =- wu(2). ^{Now if we rotate both}

the grain ^{and the nematic} by 1800 around an axis Oy

normal to the (u, no) plane, ^{we see that w _} u(2)

wu( 1 ).

Thus wu(2)

ID-u(2) ^and o.

ID-u, Q. ^{E. D.}

The most general ^vector constructed from u and _no, and even in each of them, ^{is of the form :}

where 0 is the angle (no, ^{u) and} 1(x) ^{is an} odd function of x. Inserting (II.4) ^into (II.3) ^and (II.2) ^{we find for}

the long range distortions

where _ul

u - cos 0 no is the component ^{of u in} the _xy plane. ^Note that the coefficient of 1/r ^vanishes

when ^u is parallel ^to no and also when u is normal to no (since ¹⁽⁰⁾

0). The function l(cos ^{0) has the}

dimensions of a length, ^{and we} call it the distortion

amplitude. ^For many small motion problems ^the quantity ^{of main} interest is the value 1(cos o = 1) = ll (which may be positive ^or negative). ^{We shall see}

later that for a long ^rod Il ^is usually positive ^and comparable ^{to the} length of the rod.

II .1. b Connection between distortion amplitude ^and

distortion _energy.

Consider a surface Z surrounding

the grain, ^but very distant from it. When we impose

a fixed value of 0 the grain applies ^a torque ^{l’ on the} nematic inside Z. Nematics transmit torques : ^the molecules inside X exert on the molecules outside 1 the same torque ^{r. It is} possible ^to compute ^{this tor-} que, since in the vicinity ^{of E} the distortion has a

known form (eq. II . 5), ^{and thus to relate r and} 1(cos 0).

The calculation proceeds ^{as follows : we} keep ^the grain ^{fixed in} orientation (u fixed) ^{and we rotate the} general ^{nematic direction} no by ^an angle ^b8 (around

an axis normal to u and no : ^see Fig. 4). ^The change ⁱⁿ

FIG. 4.

Distortion in

nematic induced by

^{rod :}

^is parallel

to Z at the grain ^surface.

distortion energy for the nematic inside 1 is to first order

where (i) = x, y, z, ai == a/oXi and summations are

taken over repeated ^indices. Integrating by parts ^we get ^{a volume} contribution, which vanishes automa-

tically ^{from the} equilibrium equations, ^{and a surface} contribution ; ^the integral ^{on the} grain ^{surface vanishes}

and we are left with ^an integral ^{on 1}

Since Z is ^a distant surface, ^we may replace ^bn by bno in eq. (II.7) and insert _eq. (11.5) ^{to compute} ai n j’ The result is

The integral ^is equal ^{to - 4} n, and the change ⁱⁿ

distortion _energy reduces to

In vector notation, ^we may say that the torque ^r applied by ^the grain ^{to the nematic is}

Thus, ^{to know} f (cos 0) ^is equivalent ^to knowing

the () dependent part of the distortion energy. For

instance, ^if l(cos 0) ^is positive for cos 0 > 0, ^the

distortion _energy is a minimum at f)

0 and a

maximum at 0

n/2.

II .2 APPROXIMATE CALCULATION OF THE DISTORTION AMPLITUDE AND ENERGY.

We consider rodlike parti-

cles (Lld - 10) ^{imposing a certain boundary condi-}

tion ^on the director ⁿ at the grain ^{surface. We shall}

discuss the following simple ^{cases of} boundary ^condi-

tions :

a) The molecules want to be tangential ^{to the sur-}

face and along ^one preferred direction in this surface.

We shall consider the cases where this direction is parallel ^or perpendicular ^{to the} grain ^axis.

b) Again ^{the molecules are} tangential, ^{but there}

is no preferred ^{direction in the} plane ^of contact ; the boundary conditions are degenerate. ^{The elastic} equations ⁱⁿ the bulk have then more than one solution,

and the correct one must be obtained by minimizing Felas with respect ^to the molecular orientations inside the plane ^{of contact.}

c) The molecules must be perpendicular ^{to the} grain ^surface.

In all three cases (a) (b) (c) ^we neglect ^{end effects} whenever possible. ^The general argument ^{for this is}

as follows : the lateral surface of the cylindrical grain

is

Ld and the distortion _energy (11.1) ^{due to the} regions around it is of order

since 1 Vn 1 - 1 /d in a thickness - d near the surface.

On the other hand the ends of the cylinder ^{have a}

surface d2 and contribute an energy

Thus for d ^{L end effects are indeed small.}

II. 2. a Molecules at the grain surface parallel ^to

the grain ^axis.

^{We use} coordinates X, Y, ^{Z with Z} parallel ^{to the} grain ^axis (Fig. 4) ^and try ^a variational solution of the form

This gives :

and the optimum corresponds ^to V2 ^a

0.

The boundary conditions are a

0 on the cylinder

and

= 0 at infinity. We may write

where C is the capacitance ^{of a} conducting ^{rod identi-}

cal in shape ^{to our} grain. ^In practice ^{we shall use for C}

an approximate value, corresponding ^{not to a rod}

but to an elongated ellipsoid :

Using (II.11) ^and (II.10) ^{we arrive at}

Eq. (II.13) ^{is valid even for} large 0 ; ^it violates, however, ^the parity requirement F(e)

F(rc - 0).

In fact, F(O) ^{has two branches} F1(O) ^and F2(O) ^corres- ponding ^{to two nematic} arrangements (figure 5),

and F,(O)

Fl (n - 0). ^One may pass from confor- mation (1) ^to conformation (2) by introducing ^a

disclination line. We will always stay ^{on the branch} of lowest energy if the process is effective.

FIG. 5.

Two possible determinations for the _energy of the rod

function of angle.

Note that _eq. (II.13) ^{leads to a} distortion ampli- tude 11

^{C which is} essentially proportional ^{to L.}

II.2. b n is tangential ^and perpendicular ^{to Z. -}

We restrict ourselves ^to the case 0

n/2 ^{and 0}

^0.

For 0

n/2, the parameters ^are described in

figure ^6a.

FIG. 6.

Distortion induced by

^{rod :}

^is tangential ^{to the} surface and perpendicular ^{to Z.}

n may be written

F is given by

ôF

0 leads to V2l/J = 0, ^{i. e.}

The solution of this equation ^is

where R

dl2 ^{is the} grain ^radius.

u,, and vn ^are determined by boundary conditions. For

u,, and vn ^{are the Fourier} coefficients of the function of

period 7r ^defined by (II.15) ^{i. e.}

W is then given by

This may be rewritten in the closed form

This configuration ^{has two} disclinations lines atta-

ched at the surface at the points

F is readily integrated ^to give

where a is the radius of the core of the disclination.

For 0

0, ^the geometry ^{is shown on} figure ^{7b. We}

calculate F by ^a variational method, choosing ^{as a}

trial function n :

FIG. 7.

Distortion induced by

^{rod :}

^is perpendicular ^to

the grain ^surface.

F is readily integrated ^to give

We note that FfI ^{Fl. We find that}

is large ^{and not} very sensitive to the diameter of the grain (e ^{is the base of} Neperian logarithms).

Il. 2. c n is perpendicular ^{to the} grain surface.-

We restrict ourselves to the cases 0

n/2 ^{and 0}

^0.

6

rc/2 (see Fig. 8a). ^{This case is identical to 1. b}

except ^{for a} 900 rotation of the molecules. The confi-

guration ^{has two lines attached at the surface at}

the points

The _energy F is still given by ^eq. (II.17). ^{Note that}

if the disclination line separates slightly ^{from the} grain surface, this may release ^some strain (because

of image ^force effects). ^{However we do not} expect this to alter very much Fi.

f)

0 (see Fig. 7b). ^{We calculate F} by ^a variational method, choosing ^{as a trial function n :}

.F is readily integrated ^to give

Again ^we find that Fil ^{FL and}

Conclusions.

If the grain ^surface aligns ^{the first} layer of nematic molecules surrounding it, ^{we find} that FII Fi ^{for an} elongated grain ^{and that}

where

is a numerical coefficient of the order of unity

and is not very sensitive to the diameter of the grains.

II . 3 FINITE CONCENTRATIONS OF GRAINS : TRANSITION FROM INDIVIDUAL TO COLLECTIVE BEHAVIOUR.

We now consider a dilute suspension ^{of identical} grains, ^with position _rp (distributed ^at random) ^and orientation up, ^{in a} nematic of optical axis no. For

simplicity ^{we shall assume that} misalignments ^are

small (11 > 0, _upl 1). ^The distortion bn(r) ^{is then} given by ^the following equation :

The _presence of ôn(rp) ^{in the bracket expresses the}

fact that grain (p) ^{creates no} long range distortion if it is aligned ⁱⁿ the local nematic direction no ⁺ bn(rp).

Let us now go ^{to a} continuum approximation : ^{if C}

is the number of grains per cm’ we find, ^after acting

with the operator V2 on both sides of equation (11.21) :

The solution is

Thus the effect of each grain ^{is screened out at}

distances larger ^than 1(-1. Typically ^{with C =} 10"

and 11

^{103 A we} expect ic-1 - 1 /-l.

Now let us assume that, ^{in a certain} region ^{R of the} sample (with ^linear dimensions D) ^we impose ^{a fixed}

value of ul, while out this region ^we put ui

0

(Fig. 8). ^{Will the} nematic molecule in R follow the

grain ^{direction or not ? The answer is} provided by eq.

(11.23). When D > ic-1, ^{Sn is} indeed identical to ul

FIG. 8.

Coupling between the grains and the nematic. In

region ^{of size D the} grains ^have

imposed orientation

Outside of D the grains

parallel ^to the nematic direction at rest _no. Case (a) ^KD

1 : all the inside ^{follows the} grains.

Case (b) ^xD lA:: the inside is only weakly perturbed.

in all the region ^R (except ^{for a thin} shell of thickness x-1 near the surface) : ^the grains impose their orienta- tion. In the opposite ^limit (D ic-1), ^bn « ul, i. ^e.

the nematic _response is small.

II.4 RIGHT HANDED AND LEFT HANDED GRAINS. -

The function l(cos 0) ^{is the same for a} grain ^{and for}

its mirror image : ^{the terms of order} 1 /r ^{in the} long

range distortion are not sensitive to chirality ^effects.

To display ^an interesting différence between right ^and

left handed grains ^{we must} include the terms of order

1/r2. ^{These terms will be dominant only when ce}

^0.

This ^can be achieved in two ways :

a) ^{With a} very small grain, ^the coupling energy between ^u and _no (of ^{order 4} nKl 1) may become smal- ler than kB T : ^{then u} points ^with equal probability ⁱⁿ

all directions and the _average of ce may be shown to

vanish, ^from eq. (11.4).

b) ^{With a helical rod} (polypeptide, virus, ...), equi-

librium will often correspond ^{to u} parallel ^to no, in which case

0.

Let us assume that we are in such a situation : then the only distortion term compatible with rotational invariance, ^{and with the fact that w is even} in no, is :

where fl ^{is a} pseudoscalar parameter (with ^{the dimen-}

sions of ^a square length), ^{and thus non} vanishing only

if the floating object differs from its mirror image. ^The aspect ^{of the} corresponding distortion is shown on

figure ^9.

Let us discuss now the case of a dilute solution of such objects ^{in a} nematic, ^all grains having ^{the same} chirality. ^{On a} macroscopic scale, ^{we know that such}

FIG. 9.

Distortion in

nematic induced by

^chiral impurity (i.

impurity different from its mirror image).

a system ^{behaves as a} cholesteric. We shall show that there is a simple ^relation between the pitch Le ^{of the} spiral structure, the concentration v and the parameter

fl defined above : for low

values the total rotation vector at any point ^{r is obtained} by superposition :

Going ^{to the continuum} limit, this leads to

The form of the solution is

where

is an arbitrary axis. This corresponds ^{to a}

cholesteric structure of half pitch

inversely proportional ^{to the} concentration. This type of law is indeed familiar for dilute solutions [4].

III. Magnetic grains. - III. 1 PROPERTIES OF A SINGLE (MONODOMAIN) ^PARTICLE.

^111.1. a) ^Ferroma- gnetism ^vs superparamagnetism.

^{We now} give ^to

our particle (of length L, diameter d, ^volume V) ^a magnetic ^{moment Il}

M, ^{V. We assume for} simplicity

that the moment p ^has two easy directions -parallel

and antiparallel ^{to the} grain axis, ^{and that it cannot} jump ^{from one direction to the other} (ferromagnetic grain). ^{If AE is the} energy barrier between these two

conformations, ^this requirement implies [5]

Possible sources for AE are : (oc) ^an uniaxial magne-

tocrystalline energy

(fi) ^the shape anisotropy

-

(y) surface effects (3). ^{In cases} (a) ^and (P) ^{AE is} proportional ^{to the} grain volume V, ^{and the ine-} quality (III.1) gives ^a minimum value for V. In most cases of interest here, V ^{will be} significantly larger

than this minimum.

III. l. b) Mechanical clamping. ^{- As shown in}

section II, ^the grain usually ^{tends to be} parallel ^{to the}

local nematic axis n(r). ^{To rotate the} grain materially

from + ^{n to -} n, ^we need an energy aKL. The grain

will be effectually clamped ^{in one} of the two positions provided ^that

(3) ^We

^{indebted to} the referee of the present paper for

pointing ^{out the} importance ^of (ce) ^and (y) ^{in the}

of interest.

where t is the duration of the experiment, ^and Dr ^is

the rotational diffusion constant of the grain (in ^an isotropic liquid ^of viscosity ^{similar to} that of the

nematic). This condition for mechanical clamping

may be rewritten qualitatively ^as

and it should be satisfied in most practical situations.

III. l. c) ^Coercive fields of ^a single grain.

^To

rotate u from, say, ^{n to - n} in direction, ^{we must} apply ^a strong ^{field H} along - ^{n. Two} types ^{of beha-} viour may be found : if u rotates without _any mechani- cal motion of the grain the threshold is [5]

while if the grain ^rotates materially, the threshold is

where h ^{is the} characteristic length defined in section II,

and is of order L. Taking ^the ratio of both coercive fields we find

The critical diameter dc, ^{is of order 50 to 100 Á.}

For d > de ^the dominant process is mechanical rota- tion. The various types ^{of behaviour of a} single grain ^as

a function of L and d are summarized on Figure ^10.

A typical ^{value of} Hmeeh (for d

²⁰⁰ A, M,

^10’

and l i

L) ^{would be 300 oersteds} (4). ^{Thus, with}

moderate fields in the kilooersted range, it should

FIG. 10.

Domains of mechanical and magnetical clamping

of

single domain magnetic grain ⁱⁿ

liquid crystal. ^{L and d}

are

respectively ^the length and diameter of the grains. ^The only meaningful regions correspond ^to L » ^d.

(4) ^{If the} grain (or ^the surrounding distortions) ^is large enough

to be observed optically, ^it might ^be possible ^to

the distortion amplitude f (9) ^from

study ^of 0(H) ^for

single grain ⁱⁿ

thin nematic slab.

be possible ^to align ^the grains completely. ^{We shall}

now proceed ^{to discuss} suspensions ^{which have} been

prepared ^{in such a} magnetized ^{state : we shall see}

in particular ^{that when} H « H mech, although ^indivi-

dual rotations are forbidden, cooperative rotations, involving many grains, may still occur.

III . 2 FERRONEMATICS.

III. 2. a Free energies ^and segregation effects.

^{We now consider a} suspension (with low volume fraction of grains/) ^{and assume that} locally ^all grains, in each small volume element, ^have parallel ^{moments. As shown} above, ^{this common}

direction is colinear to the director n(r) and, by ^a

suitable definition of _n, we may take it to be of the

same sign ^{as n. The} average magnetization ^{vector is}

then

and the free _energy density ^{takes the form}

The first term is the Frank elastic _energy [2] ^and

the last one is the entropy contribution for a dilute suspension ^{of non} interacting grains. ^{If we minimize} (III. 7) first with respect ^to n, ^we find that the torque

on the nematic molecules contains ^the usual elastic terms plus ^a magnetic ^{term Mn A} H. Minimization

with respect to f (keeping the number of grains fixed)

leads to :

The parameter fo ^{is obtained as usual from the}

total number of grains present ^{N :}

Eq. (III. 8) shows that f ^{may vary} strongly ^from point ^to point, ^i. e., grain segregation may occur,

provided ^that the number

be larger than 1. We call _p the segregation ^parameter.

Specific examples ^with p 1 and with _p > 1 will be discussed later. To end the discussion of free ener-

gies ^{we note that after} minimization the total free

energy Y ^{of the} sample ^becomes simply

as can be seen from _eqs. (III . 8) (III . 7). ^Here Fel ^{is the}

contribution of the Frank terms, integrated ^{over the} sample ^{volume. We shall now} apply these formulas

to some selected experimental situations.

III. 2. b Alignment ^near

^wall.

^{Consider a ferro-}

nematic occupying ^{the half} space

> 0. Assume that

on the limiting ^wall (i. ^{e. the} plane

0) ^{the mole-}

cules ^must be directed along the x direction. The sample ^{has been} prepared ^{with a field H} along x ;

later we rotate H and bring ^it along the y ^direction.

Far in the liquid, ^the magnetization will also rotate

by ^{90°. But near the wall there must be a transition} layer, ^{the thickness and} properties ^{of which we want}

to study (Fig. 11).

FIG. 11.

Distortion in

ferronematic

oriented wall.

At all points r ^{the director} n(r) will be in the (xy) plane ^{and we} may define it through ^the angle 9 bet-

ween n and the y ^{axis. For}

0, ç

n/2 ; ^for

^oo 9 --> 0.

The free energy per unit ^area of the wall is

and the equilibrium equation ^for ç has the form

It is convenient to define a reduced concentration

Çl = f/f, ^where f ^{is the} average of f over ^the sample volume, ^{and a} « cohérence length » Ç,2(H) by ^the equation

Typically for H

¹ gauss, f

10-3 , and MS =103, S2

10 microns. For the present case, eq. (111.8)

leads to

and _eq. (III.12) ^{has the first} integral

The choice of integration ^{constant in} eq. (III.15)

is such that dpjdz

^{0 for}

^oo (g/

1). Integra- ting (III.15) ^{we arrive at an} implicit ^{relation between}

p and z :

We shall discuss first the simple limiting ^case

p « ¹ (no segregation). ^Then eq. (III.16) simplifies considerably, ^{and the final result is :}

The decrease of ç ^with

^is nearly exponential,

and the thickness of the transition layer ^is equal ^to

the coherence length Ç2’

In the opposite ^limit p > 1 segregation ^effects

become important ; ^the general ^{aspect of} g(z) ^and

of gl(z) ^{is shown on} figure ^{12. At} very large ^distances

from the wall, qJ(z) again decreases like

FIG. 12.

Structure of the depletion layer (p

1) ^{for the} geometry of figure ^12.

However, ^{this law does not} hold down to Z ~2

and the transition region ^is larger ^than Ç2’ Its thick-

ness may be estimated from the initial slope :

where we have used _eq. (III.15, 14). ^{In the} high ^field

limit (p > 1) ^this characteristic length ^becomes

where

= fl V ^{is the} average grain concentration in the bulk. The thickness ô is independent ^{of the} magni-

tude of H, ^{and is} typically of order 10 microns. In

a depth ^{b near the} wall, ^the grain concentration is

strongly ^reduced (flf = e-P) : ^{thus a} homogeneous

field of suitable orientation will create a depletion layer ^{near the wall. The work} required ^to expel ^the grains ^{from the} layer ^is (per ^{unit wall} area) ^{of order} ckBTb (ckBTô being the osmotic pressure), ^{while the}

torsion _energy is of order K22blb2. Minimizing ^the

sum, ^{one recovers} eq. (III.19), ^{except for numerical}

coefficients. It would be of great ^{interest to} try ^and observe these depletion layers by optical ^means.

III. 2. c Ferronematic slab in reversed fields : ^insta- bility threshold.

We consider a thin slab (thickness

D ~ 10 microns) ^{of our nematic} suspension, ^enclosed

between two glass ^walls polished according ^{to the}

method of Chatelain [6]. Both walls are polished along

the same direction (x) : ^{near them,} the director ⁿ

must be colinear to x. We call

the direction normal to the walls (Fig. 13). ^{We assume that the} sample ^has been prepared ^{with a} large ^field applied along ^the negative x ^direction (giving full saturation and a

uniform grain concentration). ^{Now we reverse the} field, ^and go ^to small positive values of H. The initial

configuration ^is metastable, ^{and becomes} completely

unstable at a certain threshold field H, ^{which we} shall now derive.

FIG. 13.

Ferronematic slab in reversed field.

Consider a small deviation from the original ^arran- gement, ^{where the director n has the} following ^struc-

ture :

Inserting this form into eq. (III.7) ^one gets, ^{for the} free energy per unit wall ^area

(Note that this result is the same with or without

segregation effects.) ^{We expand} ny and nZ in Fourier series matching ^the boundary conditions at both walls :

In terms of the u’ ^{s and v’ s the free} energy becomes

Instability ^{occurs when one the terms becomes} nega-

tive ; ^this happens first for n

1. There are two possi- ble threshold fields

The physical ^{threshold Hl is} the smallest of the two. Taking ^M

fMs

¹ gauss, D

10 microns and K ~ 10-6 we have Hl 1’-1 ¹⁰ gauss. Thus, ^even

for rather small samples, ^a coupled rotation of _many

grains ^{is able to} relax the magnetic ^{moment much}

more efhciently ^than individual rotations : let us

compare Hl ^to the coercive field for one grain Hmech (eq. 111.3). Neglecting all numerical coefficients we

find

where K - 1 is the ^« screening » length introduced in eq. (II.22). Eq. (III.23) shows that collective behavior is observable provided ^{that xD} > 1, ⁱⁿ exact agree- ment with ^our conclusions of section II.

III. 2. d Behavior above threshold.

This part ^of the problem ^{is more} delicate, for various physical

reasons. Here we shall restrict our attention to one

typical case, defined by ^the following assumptions :

-

Segregation ^{effects are} negligible (p 1).

- All elastic constants are equal : ^{in this case it}

is plausible ^{to assume that there is no} twist, ^{i. e. that} the director n is always ^{in the} (x, z) plane. ^{We shall} specify ^its orientation through ^the angle g(z) ^between

n and H.

- No disclination line is allowed (we ^{come back}

to this point ^{in the next} paragraph).

With these assumptions ^the equilibrium equation

retains the form (111.12) (with ^{a constant} f). Eq.

(III.15) ^is replaced by

where _CPm is the deflexion angle ^at the middle of the slab. The solution cp(z) ^{may be expressed in terms of} elliptic functions. The boundary ^condition

leads to an implicit equation ^for qJm

Our definition of the elliptic integrals ^{such as F}

follows ref. [7].

Eq. (III.25) ^{has a solution} only ^for H > Hl.

The spatial configuration of the molecules for various H values above Hl, ^{are shown on} figure ^14.

As soon as H > Hl ^{the bulk of the} nematic is _magne- tized parallel ^to H, ^{and there are} only ^{two thin} regions

of thickness ç ^{near the walls} where the magnetization

is mainly ^{in the} wrong direction. It is also of interest to compute ^the average magnetization

A plot ^of Mav/ M ^{versus H for} this ideal case is shown on figure ^15.

FIG. 14.

Spatial configuration of the molecules for various H values.

FIG. 15.

Magnetization

^of

ferronematic slab.

III . 2. ^e Possible effects of disclinations.

The con-

formations which we have described above, ^{with a} magnetized ^{nematic in a reversed field, are meta-}

stable. They might ^{be disrupted} by the nucleation and motion of disclination lines [8]. We shall discuss here

some specific ^{features of these lines} occurring ^when magnetic grains ^are present.

As pointed ^out earlier, ^{in most cases the} average

magnetization M(r) ^is parallel ^{to the} local nematic

axis (i. ^e. parallel ^or antiparallel ^to the director n).

Consider now the disclination loop Lo ^{shown on} figure ^{16. We assume that} Lo is of rank 1, ^i. e., if we

follow a circuit such as C, starting ^{from a director n}

at point ^{1 we obtain a} director - n at the same point after one turn.

FIG. 16.

Disclination line Lo and mismatch surface Zo ⁱⁿ

ferronematic.

This shows that M cannot be constantly parallel ^to and of the same sign ^{as n : there must be at least one}

discontinuous point ^{D on C. The locus of these} points

is ^a surface of discontinuity Zo, ^bounded by Lo. ^When

we cross £0, ^{we find an} abrupt 1800 rotation of the vector M. An example ^{is shown on} figure ¹⁷ (5).

FIG. 17.

An example of mismatch surface Zo showing ^the discontinuity ^of magnetization vectors. For this example, Z’o

carries magnetic poles (of negative sign).

Of course, the discontinuity ^{is not} infinitely sharp,

for various reasons :

a) ^{If we had some direct} (e. ^g. dipolar) interaction between grains, strong enough ^to compete with the local nematic alignment, they could build _up something

reminiscent of a Bloch wall. But, ⁱⁿ the present paper,

we are concerned with dilute colloids, ^{for which such}

interactions are negligible.

b) ^{A more} important ^{effect is} translational diffusion

of the grains : ^this implies ^{widths of order} .J Dt

where D is a grain ^diffusion coefhcient, ^{and t the}

(5) ^{A similar} discussion, linking ^{walls and} loops of rotation

disclinations, ^has already been carried out for solid state ferro-

magnets ^and helimagnets by ^{M. Kleman} (Phil. Mag., ^{to be}

published).

duration of the experiment. ^il

10-8 cm’/s ^and

t

1 s would correspond ^to àDt

¹ micron, ^a very sizeable thickness.

Let us assume for the moment that the surfaces of

discontinuity ^are sharp : ^{The situation} is then similar to that of imperfect translation dislocations in ordered

alloys, ^{as has been} pointed ^{out to us} by ^{J. Friedel.}

(The surface of magnetic mismatch is equivalent ^to

a surface of stacking fault for translation dislocations.)

But in our case, there is essentially ^{no « mismatch}

energy associated with £,.

Thus we expect the disclination loops ^{to be} quite mobile, just ^{as in a} conventional nematic.

When a loop ^{L moves, it sweeps a surface r, as}

shown on figure 18, ^and the surface of discontinuity

becomes 10 ^{+ h} (6). ^{Thus a} moving loop ^{leaves a}

« trail » behind it.

FrG. 18.

Motion of

disclination loop ⁱⁿ

ferronematic.

It is difficult to predict what could be the nucleation and growth processes for such lines in a nematic slab under reversed fields. Let us, for simplicity, ^restrict

our attention to cases where segregation ^is negligible (p 1). ^{As noted} earlier, ^if H > H,, ^the magnetic

coherence length ç(H) ^{is small} compared ^{to the sam-} ple thickness, ^{and the} initial conformation near one

wall is that of figure ^19a. Starting ^from this, ^one possi- (6) Note however that, ^{in the} presence of

external field H

magnetic stresses may distort the surface of discontinuity.

FIG. 19.

Nematic conformation in reversed fields (a) ^without

disclination b) ^with

disclination loop ^{L in}

plane parallel ^to the wall and normal to the sheet.

ble sequence of ^events would be the following : ^a loop ^{nucleates at the} surface, ^{moves towards the inside}

of the sample up ^{to a} thickness of order ç, ^{and simul-} taneously expands ^{in the slab} plane (Fig. 19b). ^If

all the wall surface is swept by ^such loops, ^{we arrive}

at a situation where the nematic alignment is uniform.

But the magnetization is still in the _wrong direction in ^a sheath of thickness j ^{below each wall} (Fig. 20).

These sheaths might ^{either smear out} by translational

diffusion, ^{or show} further instabilities due to the

magnetic ^stresses.

FIG. 20.

Smearing ^{out of the} magnetization profile by ^diffu-

sion for

ferronematic, in reversed fields, ^after sweeping by disclination lines (limit p 1).

This scheme is highly conjectural (clearly ^direct optical observations would be desirable). But it is worth while to observe that with such a scheme, ^and

if the sheaths smear out by diffusion, ^the magnitude

of the sample ^{moment in the final state is} smaller than the saturation moment, ^as is obvious ^on figure ^{20. To}

restore the full saturation moment, ^one might require IH 11"’>.1 Hm,,,-h, i. ^{e. fields} large enough ^{to rotate the}

individual grains.

III. 3 FERROCHOLESTERICS.

III. 3. a Definition of

the model.

We call « ferrocholesteric

a cholesteric suspension ^{where the} magnetization ^{vector M} spirals

in phase ^{with the local} optical ^{axis. If we call}

^the