Viscosity and moving dislocations in colloidal crystals

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Viscosity and moving dislocations in colloidal crystals

M. Jorand, A.-J. Koch, F. Rothen

To cite this version:

M. Jorand, A.-J. Koch, F. Rothen. Viscosity and moving dislocations in colloidal crystals. Journal de

Physique, 1986, 47 (2), pp.217-227. �10.1051/jphys:01986004702021700�. �jpa-00210198�

Viscosity ^and moving dislocations in colloidal crystals

M. Jorand, A.-J. Koch and F. Rothen

Institut de Physique Expérimentale, Université de Lausanne, 1015 Lausanne-Dorigny, Switzerland (Reçu ^{le 2 mai} 1985, accepté sous forme définitive ^{le 8 octobre} 1985)

Résumé.

Nous étudions le mouvement d’une dislocation vis dans les cristaux colloidaux et appliquons ^ce

calcul à un écoulement de Couette. On en déduit une relation analytique pour la viscosité macroscopique ~ ^en

fonction d’une viscosité intermédiaire ~, de la densité n de dislocation et du cisaillement. Une modification des contraintes internes au voisinage ^{du coeur de la} dislocation, lorsque ^{sa vitesse} change, ^{entraîne une diminution} de ~ lorsque le cisaillement externe augmente.

Abstract.

We study the motion of a screw dislocation in colloidal crystals ^and apply the results to a Couette flow. We get ^an analytic expression ^{for the} macroscopic viscosity n ^{as a} function of an intermediate viscosity ~,

of the dislocation density ⁿ and of the shear. A modification of internal stress near the dislocation _core, when its velocity changes, implies ^a diminution of ~ when the external shear rises.

Classification Physics ^Abstracts

47.55K - 82.70

61.70G

Introduction.

There has been a great deal of interest in the physics

of colloidal suspensions [1] and among them, ^the colloidal crystals present very peculiar physical pro-

perties ^{such as} phase transitions under shear [2].

Colloidal crystals [2] (c.c.) ^are monodisperse ^colloids

in which the colloidal particles, through dissociation reaction in water, ^have strong ^electric charge. (Ze N

1000 e for a typical ^{diameter ø ~ 0.1} pm). ^Under

well defined conditions (density ^of particles, charge,.

electrolyte concentration), ^{the most} interesting phe-

nomenon resulting ^{from the Coulomb} interaction between particles ^{is their} ordering ^{into a} crystalline lattice with a typical ^{lattice constant a N} 1 J.1m.

As any solid, ^{these c.c. have a} finite elastic modulus

(E) whose order of magnitude ^can be estimated [3]

by multiplying ^{U, the} energy of interaction between two particles by ^{n, the} density ^of particles :

For a charge ^{Ze - 10’ e and an} interparticle ^dis-

tance a - 1 _pm, the Coulomb interaction

- er ^gives

U N 10 eV with a dielectric constant c - 100. One finds : E - 1 _a3 ^{U - 1 dyn Y cm-’ while in a usual}

solid E - 1012 dyn ^{cm- 2.}

The above estimate has been confirmed by ^various experiments [4-6]. ^{Such a} low value of the elastic modulus will have _very interesting consequences,

especially for the flow and melting of these colloidal

crystals.

In this paper, ^we consider a model of dislocation motions to calculate the behaviour of the ^macrosco-

pic viscosity ^{of c.c.} flowing ^under applied ^{shear. In}

the first section we review the basic experiments studying ^{the flow} properties ^{of c.c. ; in the second sec-}

tion the model is presented ^{and discussed and in the}

third part, numerical results are compared ^{to the} experimental ^results.

1. Experimental ^review.

The most appropriate geometry ^to study ^flow pro-

perties ^{of c.c.} is the Couette flow (Fig. 1) (two ^coaxial cylinders with different angular velocities). ^{In this} geometry, ^{Mitaku et al.} [7] ^{have measured the shear}

rate Vij ^{as a function} of the shear stress Gij ^{for the flow} of ordered c.c. Typical curves are given in references

[7] ^and [8] ^{but the} qualitative ^{behaviour is} reproduced

here (Fig. 2).

The main result is that, in the ordered state, the ^c.c.

behave like a non-Newtonian fluid, ^whose viscosity

diminishes with increasing ^shear (pseudo-plactic

fluid [9]).

If we consider a stress-strain relation [7] of the form : G r8

nVre ^{+ K, where K is the} yield stress, from the measurements of Mitaku et al. ^{we can} draw the macroscopic viscosity -q ^{as a} function of the angular velocity ^w of the inner cylinder, ^{the outer} being ^at

rest (this is shown in Fig. 3).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004702021700

Fig. ^1.

Geometry ^{of a} Couette flow experiment.

shear stress (dyn.cni 2)

Fig. ^3.

Diagram ^{of the} macroscopic viscosity t-I ^vs.

angular velocity ^w (w ~ ^shear rate).

In the experiments [7,8], V,,, > - ^{6 co} (see Eq. (36)).

At low shear rate (cm - ^10-3 s-1) ^this viscosity n ^is

four orders of magnitude greater than that of water

(17H,O - ^{10- 2 p)} and decreases to - 10 _p for m - 10-1 s-1.

For much higher shear rates, Lindsay and Chaikin

[10] ^{observe a shear} melting transition between ordered and disordered c.c. For their sample ^{this occurs at a}

critical shear rate ( V’,Io > - ^{60 s- l. The} viscosity ^is

then of the order 5 ^x 10- 2 _p and above this melting

transition the colloid is non-Newtonian with a vis-

cosity ranging (1-10) ’ ^10-1 p.

These results are in qualitative agreement ^{with the} results obtained ^on polymeric ^fluids [11], ^{but from a}

theoretical point ^{of view, a} statistical approach ⁱⁿ

our case is more difficult since we must add a Coulomb interaction between particles.

Another interesting experiment ^{has been} perform-

ed by ^{Chaikin et al.} [ 12] ^{in a Couette} geometry with

a very small _gap between the cylinders.

Under low shear rate, the ordered c.c. flow and this flow can be described by dislocation motion.

Indeed, ^{in this} experiment ^the dislocations form walls

(Fig. 4) and these have been observed through ^the modifying light scattering properties ^{of the} sample.

(The ^{existence of} these walls is not surprising ^since they correspond ^{to a stable} equilibrium ^of grain boundaries) (see ^{for instance Ref.} [13]).

q Fig ^2.

Qualitative behaviour of shear rate vs. shear stress

(from ^paper by ^{Mitaku et al.} [7] Fig. ^5, 130uM ^{in kCl}

concentration). Dotted line : Newtonian fluid; ^continuous

line : ordered colloidal crystals; ^main characteristics of the

sample : volume fraction - 12 % ; shear modulus

p - 102 dyn cm- 2 ; ^{lattice constant a - 3 x} 10- 5 cm.

Fig. ^4.

^Periodic stripes ^of grain boundaries (wall edge dislocations) in Couette flow experiment [12] (a - 100).

Another experiment of shear-induced melting by

Ackerson and Clark [14] ^{has been} interpreted ^{as the} slipping ^of two-dimensional hcp layers but in this work we assume that the whole flow is due to disloca- tions. Moreover, ^we restrict ourselves to very low shear rate regime ^{and from the above two} experiments

we calculate I by studying ^the dislocation motion in this regime.

2. Theoretical approach.

We first present ^the general ^idea leading ^{to an esti-}

mate of the macroscopic viscosity ^{of these c.c., then}

we discuss the motion of ^a single dislocation through

the theoretical description ^{of the c.c.} and results introduce these in the general scheme of the first part ^{in order to} calculate this viscosity.

2.1 How To CALCULATE fi. ^{- From now} on, ^we describe our c.c. in the Couette geometry and, ^as figure ⁴ suggests, ^{we treat} straight dislocations whose

axes are parallel ^to the z-axis (Fig. 1). ^{So we have a}

two-dimensional problem.

There are two ways ^to discuss the motion of c.c.

under shear flow :

1) ^The macroscopic point ^{of view treats the c.c.}

like a fluid of macroscopic viscosity I without refer-

ring ^{to what} happens ^{to the structure} inside this fluid. As we saw in section 1, ^{the order of} magnitude of i ranges from 103 _p to 10-2 _p, n being ^shear depen-

dent

2) ^The intermediate point ^of view considers a

scale length ^on the order of the interdislocation dis- tance. At such a scale, ^we treat discrete dislocations in

a continuous elastic medium. Submitted to an exter- nal shear, ^the dislocations move and their displace-

ment fields induce hydrodynamical ^{currents which are}

characterized by ^an intermediate viscosity q.

For an isolated moving dislocation, ^{we have a}

small local displacement of the elastic medium and the

viscosity associated with this motion can be compar- ed to the viscosity appearing ^{in shear waves} experi-

ments [5]. ^{For a} typical sample (u - ¹⁰² dyn cm-’,

a - 3 x 10- 5 cm) ^we get 11 ’" ^0.1 p.

One could describe the ^c.c. in a third _way by ^consi- dering lengths smaller than a and starting ^{from the}

fundamental level of a pure liquid ^{with a viscosity}

110 ’" 11820 ’" 10-2 _p.

Then we have three scale lengths and three kinds of viscosity ^{and we} ought ^{to start from} 110 ^to arrive at to have a continuous description ^{of these c.c. But}

in this work, ^{we will} skip ^{the first} step ^and just ^find

a relation between n ^and 11.

For this purpose ^we evaluate the _power dissipated by the Couette flow through the two descriptions

and equal the final results. Let P

P(w, -j7 ^{be the} dissipated power in the macroscopic description ^and Pi(V’, n) the power dissipated by ^an individual mov-

ing dislocation (Yo is the dislocation velocity). ^More-

over, if we assume that there is no hydrodynamic ^inter-

action between dislocations [15], ^{we can write :}

If we still find a relation between the dislocation velo- cities and the shear rate, ^{we will} get ^from (1) ^{a relation}

for this macroscopic viscosity :

for a given sample ^and geometry.

2.2 THEORETICAL DESCRIPTION OF c.c.

As figure ⁴

suggests [12], ^we ought ^{to consider} edge dislocations but these are much more difficult to treat than screw

dislocations, ^since they ^{contain a non-zero} displace-

ment divergence which describes compression ^or

dilatation around them. However, by symmetry, this main difference disappears ^{if we sum} the contri- butions of all dislocations in the wall [16], ^{and for}

and order of magnitude, ^the power dissipated by ^a moving ^{wall of N} edge dislocations will be obtained

by summing ^this power ^over N uniformly distributed

screw dislocations submitted to the same constraint.

(These ^screw dislocations do not reproduce ^the geo- metry ^{of the c.c.} but this is not important ^for evaluating

the _power dissipated by ^their motion).

2.2. 1 Displacement dislocation.

If we ^field confine ^around our c.c. ^{a moving} between ^screw two infinite parallel plates and consider the ideal case of

only ^{one screw} dislocation in a perfect surrounding

medium (Fig. 5).

In Cartesian coordinates (x, y, z), ^we take the Bur- gers ^vector b

(0, 0, b) ^{with b -} a ; the direction of the dislocation is defined by ^{a unit} vector = (0, 0,

-

1) tangent ^{to the} dislocation line : then we-consider

a left handed screw dislocation [13] (b · 0).

Fig. ^5.

Moving dislocation between parallel plates 4

^{sense of the} dislocation

b - 0 : left ^{handed screw}

b

Burgers vector dislocation.

If we move uniformly ^{the upper and lower} plates ⁱⁿ

-

z and + z directions respectively, ^the resulting ^cons-

tant shear induces the motion of the dislocation in the

plane (x, z) ^{at a constant} velocity Yo and in the + x

direction [13, § 3, 6].

The problem ^{now is to find the} displacement ^field

around such a dislocation and for this we need the

hydro dynamical equations describing ^the dynamics

of colloidal crystals.

These have been presented ^elsewhere [17] ^{and for}

an incompressible deformation (as it is for a screw

dislocation) ^{and slow} motion, ^the equation ^{of motion}

has the following ^{form :}

where U is the local displacement of the continuous elastic medium,

p is the shear modulus [16, 5], and ’1 ^{is an effective}

transverse viscosity [18, 5] p being ^an average density

of the suspension ( N ¹ g em - 3).

The velocity ^V

au/at describes the hydrodyna-

mical currents induced by the motion of the disloca-

tion, and since these currents are characterized by ^a viscosity ’1, the power dissipated by them is sim-

ply [18] :

where

For the particular problem ^{we now} consider, ^{we look}

for a solution of (2) .in the form

where Vo ^{is the} dislocation velocity.

Introducing (5) ^into (2), ^{we have :}

or

with

is the velocity ^{of shear waves} [16]

For motions such that Vo CS, ^we have /32 ¹

and we can neglect the inertia term of (7), ^{which is} simply ^written

Through ^this simplification, ^{we donnot} consider the energy radiated by ^the dislocation as it moves [19],

but naturally ^we ought ^to verify ^{the above} assumption

for the experiment ^{we consider.}

From the relation V

aU ^and equation (5), t ) ^{we can}

still write the _power dissipated ^{as :}

We can now look for ^a solution of (12) ^{but since} U(x’, y) characterized the displacement held around

a dislocation, ^{we must} add the boundary ^condi-

tion [13] :

where the integral ^{is taken over a} circle of radius r

around the dislocation axis. (We ^{do not now consider}

the exact form of f(r)).

From equation (14), ^{we see that} Uz ^{is a} multivalued function but for mathematical simplicity, ^{we can also}

consider it as a uniform but discontinuous function

[16, § 27].

Suppose ^{there is a} discontinuity ^{on the} negative

x-axis. The boundary ^{condition will be written} [13] :

lim Uz(x’, s) - Uz(x’, - s)

6Uz = b8( - x’)f(x’)

e-0

with

For a usual crystal [13], f(x’)

1. But for c.c., the motion of a dislocation is slackened by ^{the visco-} sity ^{of the} liquid which slows down the displacements

of the elastic medium. One can study ^{this more} quanti- tatively by considering ^the dynamics ^{of the disconti-} nuity ðUz appearing ^{behind a} moving ^{screw dis-}

location. Let us decompose 6 U_, according ^to figure ^5a

Fig. ^5a. - Discontinuity ð Uz given by ^{a classical} part ð U.’ ^{and an} additional effect 6 Uz ^{due to the} viscosity.

where 6Ufl ^{is the} discontinuity appearing ^{behind a}

« classical » screw dislocation : 6 U,,,o

bO(Vo t

x), 6U/ ^{is a} correction of the discontinuity ^{due to the} viscosity, ^and 6 U,, ^and 6 U/ ^should satisfy ^the following

conditions

a) if the dislocation velocity Vo goes to zero,

ðUz1 ^should decrease since the viscosity ^effect decreases;

b) far behind the dislocation (where ^{all relative} displacements ^{of the} elastic medium have been

damped) ^{one has} ðUz1

^0;

c) in front of the dislocation (where x - Vo t > 0) ðUz

^0.

The simplest nontrivial equation ^{one can} imagine

to describe the dynamics of ð U zl ^{is :}

T is a relaxation time which can be roughly ^estimated

with the help ^{of the two} effects in competition : ^the elasticity ^{and the} viscosity

The solution of (17) which satisfies all the above- mentioned requirements ^{is :}

(the singularity b(Yo ^t

x) appearing ⁱⁿ (17) ^{is not}

relevant since it is contained in the dislocation core).

The discontinuity (16) ^{is then} given by :

where the length ’1" 0 = ^{a is} the interaction _range of the liquid ^on the dislocation. From now on one

writes ^x for x - Vo t ^and (20) ^writes

Then we have to solve equation (12) ^with boundary

condition (21), ^this exercice is done in Appendix A, and we get ^{an exact solution :}

where E1 z) ^{is the} exponential-integral [20] ^defined by :

with the properties :

and for z large :

(Fig. ^{6 describes} the effects of this motion and the

discontinuity (21) ^of Uz).

Fig. ^6.

^{Draft of a} moving ^screw dislocation, showing

a vanishing discontinuity ^{of the} displacement ^field Uz ^{at the} origin (dotted discontinuity

pure elastic case). ^Two nearly points A(xo, yo ⁺ s) ^and B(xo, Yo - c) ^{after the} passing ^{of the} dislocation will be at A’(xo, yo ^{+ a,}

b/2)

and B’(xo, Yo - B, b/2) ^with opposite velocities. Summing

this effect over all dislocations reproduces ^{the flow of}

c.c. under constant shear.

From (23) ^{we see} that, within the limit a - 0, equation (22) gives :

a standard result in the usual elasticity theory [ 13].

From (22) ^we get ^an expression ^for V z ^{as :}

with

This velocity describes the hydrodynamic ^currents

Fig. ^{6a, b.}

Comparison ^{of the} discontinuity ^of Uz ⁱⁿ

usual crystal (a) ^{and colloidal} crystal (b) (lattice spacing :

b

10-4 em).

I

Fig. ^6c.

^Draft showing ^the discontinuity ^of Yz

ðtUz

on x 0.

induced by dislocation motion and ^as figure ^6c shows, Vz ^is discontinuous _{on y}

0 for x 0.

But this discontinuity ^is meaningless ^{in a discrete}

lattice and we interpret ^{it as a} strong change ⁱⁿ velocity ^{over a} length ^b.

If we define Vz ^{such that}

where m=x+i ^.b

where ffi = x a + i 2 b oc ^{oc IX}

Then we have a continuous velocity giving ^a

viscous stress

were 110 ^{is the} viscosity of the fluid in a band of width b and it is thus equal ^to 11820 ’" 10-2 _p.

Since , b - ^{110’ we have} (11 ’" 62 and for sim p li-

a

city, ^{we then assume}

on the whole domain and forget ^the singularity on a .

2.2.2 Power dissipated by ^a moving dislocation. -

Introducing (24) ^into (13), ^{we find the} hydrodynamic

power dissipated by ^a moving ^screw dislocation :

and L is the dislocation length.

The lower integration ^bound corresponds ^{to the}

extent of the ^core of the dislocation and we can

determine the curve C(xc, Yc) by assuming ^that

inside it, the elastic stress Q is greater ^{than a} critical

stress a,. We ^assume this critical stress to correspond

to the « theoretical elastic limit » [21] whose order of magnitude ^is

Introducing (22) ^{into the} expression of the elastic

stress

we get

and from this the curve C(xc, Yc) ^is given by

with

For a vanishing velocity Yo ^{we find a circular core}

whose radius rc ’" b is the admitted classical value for a pure elastic medium. Some curves C(xc, Yc)

have been drawn in figure ^{7 for} various velocities

Fig. ^7.

^{Core section of a} moving dislocation at various velocities [cm S-1].

and we see that above a given value, ^{the core is vanish-} ingly small; yet it doesnot make sense to consider

a core with r, b, ^{since we treat} the elastic medium

as a continuum. Therefore it seems natural at this

point ^{to add a} restriction ^on C(xc’ Yc) by requiring :

Another contribution to this dissipated power ^comes from the phonon interaction and periodic ^structure

of the elastic medium as in any usual crystal. ^This

contribution has been estimated by ^Leibfried [22]

and Lothe [23] ^{and an order of} magnitude ^is given

in reference [ 13], § 7 :

where WT ^{is the} density of thermal vibrational _energy

An estimate of (25) within the limit a -+ 0 gives :

and the ratio

r -of --

r ^L-

_105.

and the ratio ^- 105 .

P’ 4 1t pb x 10-

’~

As a consequence, the power dissipated by ^a moving

dislocation is essentially given by ^Phya..

2.3 MACROSCOPIC DESCRIPTION OF c.c.

As we said in section 2.1, ^{we will} consider ^{the c.c.} like a fluid of viscosity I ^{submitted to a} Couette flow. The

angular velocity of the inner cylinder ^{is w and since}

RZ R1 ¹ (Fig. 1) ^{in the} experiment ^{of Mitaku} R1

et al. [7, 8] ^{we can} suppose ^{a constant} shear between

cylinders ^{and thus for a} given ^{w, we assume} that I ^is

constant.

We can then solve the Navier-Stokes equations [18]

for such a flow, ^and assuming cylindrical symmetry and azimuthal flow, ^{we find the} well-known solution

2 / n2B

From (32) ^we get ^the macroscopic dissipated ^po-

wer [18]

Before comparing (25) ^and (33) ^{we must find a rela-}

tion between the angular velocity ^{m and} dislocation

velocity V o.

In the limit of a plane ^motion (Ri, R2 -> ^{oo and} R2 - R,

Cte) ^{we find}

dislocation density . (

For the cylindrical ^{case with} R2 - Ri 1, ^{we can} R

write :

where n is the ^mean dislocation density

As said in section 1, ^{we consider} very low shear rates

and consequently ^we can assume that the density ^of

dislocations is constant. This is certainly ^not right

when we approach the critical shear rate of melting

transition [10] ^{but is} fairly ^{correct in our case. So} equation (36) ^{means that, for a} given w, the disloca- tions have the ^{same mean} velocity Vo, which is rather

j? 2013 R1

correct since R2 - Ri ^{R 1 1. (W e are near the plane}

case.) ^From this, ^{the result} (25) ^for plane motion will be a good approximation ^{of the} cylindrical problem.

2.4 DETERMINATION OF THE MACROSCOPIC VISCOSITY.

-

From the above discussion, ^{we introduce} (25)

and (33) ^into equation (1) and write :

or

with

If we introduce x’ = x and y’ - y , ^we get

a a

From (39) ^{we see that the inferior} integration ^bound

determines the behaviour of ti. Within the limit of small velocities such as b > a ^{1, and for the} limiting plane ^{case, we} find

in perfect agreement ^{with our} previous ^result [24].

With in this limit, ^the viscosity I ^{does not} depend

on the shear rate m but decreases as the dislocation

density increases, since the flow is facilitated.

2. 5 DISCUSSION.

We can understand the behaviour

of I qualitatively by observing ^from (37)

For small velocities Vo ^{with a} b, the characteristic

length ^{is b and we have}

So in this domain ?-I - - 172and _nb ^{we find the result (40).}

But for velocities such that ex >> b(VO >> ^{10 -2 cm} s- 1) ^the characteristic length ^{is ex and we have}

which implies

giving ^a decreasing of I ^for increasing shear, ^since - Vol

In order to compare ^our result quantitatively ^with

the experimental ^{one, we} performed ^the integral (39) numerically. The result is presented ^{in the third}

section.

3. Numerical results.

Our purpose is ^{not to} fit the experimental ^{results with}

our calculation, since ^we do not exactly ^{know the}

values of the physical parameters (appearing ⁱⁿ

Eq. (39)) characterizing ^a given sample. ^{We have}

nevertheless considered Mitaku’s experiments [7, 8],

to fix these values; ^this provides ^{an order} of magnitude.

If we consider Chaikin’s experiment [12], ^the density

of dislocations is given by ^the minimum value neces-

sary ^to reproduce ^the geometry ^{of the} sample. ^We

then get [13] :

With this value of n, ^{we can} estimate the dislocation

velocity through equation (35) with Vr8 ^{)max ~} 10-1 s-1 and find Võax ’" ^{10-1 cm s-1.}

This is much smaller than the sound velocity

and vindicates the simplification ^of equation (7).

We have plotted ^the theoretical (continuous line)

and experimental (dotted line) ^values of I ^{as a func-}

tion of w, this is shown in figure ^8.

As in the experiment, ^{we observe a continuous}

decrease of In ^for increasing ^{co. For m 10-3} s-1,

the viscosity ^{is close to a constant value} given by equation (40) (t-l - ²⁵⁰ p) ^{and it decreases} to tj - 3 p for w - 10-1 S-1.

As said above, the difference in magnitude ^{is not} significant. ^{It is more} instructive to consider the shear dependence ^{of the} viscosity ^{and we} easily ^see

a disagreement ^{on this} point.

For the experimental ^{curve, we have a} power law in the form :

with d - 0.75 for the reported experiment, ^{while for}

Fig. ^8.

Comparison ^of experimental (dotted line) ^and

theoretical (continuous line) ^results of 1-7 ^{vs. co.} Experimental

value of the parameters [8] :

n 11

the theoretical curve, ^we have a logarithmic depen-

dence :

for

and

This main difference between experimental ^{and theo-}

retical results _may be explained by dimensionality

effects. By computer simulation [11] of non-Newto- nian simple ^fluids undergoing planar ^Couette flow,

it appears that in 2-dim. the viscosity ^{of these fluids}

is given by q(y)

^A log (By) ^where y is the shear rate, while the result in 3-dim. is

But our theoretical approach ^{of the} problem ^{is 2-}

dimensional since we considered straight parallel

dislocations and we must not be surprised by ^this logarithmic dependence.

4. Concluding ^remarks.

If the quantitative disagreement ^between experiment

and theory ^is really ^{due to} dimensional effects, ^our

task is quite hopeless, ^{due to the} great mathematical difficulties to treat 3-dimensional dislocations. To

improve ^our results the first thing ^{we could do is to}

consider straight edge dislocations (as they ^are really)

but we are convinced that the order of magnitude ^will

not change. ^{In the same} way the assumption ^{of the}

dislocations independence ^{seems to be correct for}

such a density (in ^{a wall of} edge dislocations [12],

their separation ^is

^{10 b} [15]), ^{but if we decrease}

the radius of cylinders, ^the density rises and the hydro- dynamic interaction between dislocations would have

important ^effects.

Another _way to improve ^{our results would be to}

consider a varying density ^of dislocations but, ^{as said}

in section 2. 3, ^{this does not seem to be} right ^{for such}

a low shear rate. Whatever the detail of the model

developed ^here, the flow of ordered suspensions ⁱⁿ

terms of the dislocation motion gives interesting

results.

The main point ^{in this work is related to this} length

a = ’1 V 0 ^{which for a} > b implies ^a deep change ⁱⁿ

JL

the velocity dependence ^{y of the viscous} strain - yo a

around the dislocation axis by ^{reference to the} quasi-

static case - b V0. ^{As a direct} consequence, ^q

^we observe a decrease of the macroscopic viscosity ^{as the}

shear raises. It would be interesting ^to consider the behaviour of ^{c.c. near} shear melting by extending ^this

model to the consideration of higher shear rates, with a varying density of dislocations.

Acknowledgments.

We are indebted to W. Benoit for a fruitful discussion

on the dislocation theory ^{and we thank B.} Pansu, E. Dubois-Violette and P. Pieranski for their helpful

comments.

We thank the referees for their instructive remarks.

Appendix ^A.

We have to solve equation (12)

where Uz ^{is a uniform but} discontinuous function ^on the negative x-axis. It must then satisfy ^the boundary

condition (21) :

with

Following Landau’s method (16, § 27), ^since Uz ^is discontinuous, its derivatives and then the stress aij

are singular. ^But physically, ^{it is} meaningless ^{to have}

such singularities ^of _Uij and in order to cancel them,

we add a fictitious force f(sing.) such as

From (A. 1) ^{we have}

with

where

6 (y) is the Dirac distribution . The singular part ^of _a,j ^{is then} given by

with a singular ^force

We now introduce (A. 7) ⁱⁿ (A. 3) ^to get ^with (A. 4)

We can solve (A. 8) by introducing Green’s function

G(x, y) satisfying

.

If we define the Fourier transform h(p, q) ^of h(x, y) by

from (A. 9), ^{we find}

In the same way, if H(x, y)

^b 6’(y) 0(- x) ^we

have [25]

From the definition of Green functions [26] ^{and the}

Fourier transform of convolution products [25],

we can write

It is a standard exercice of residue calculus to get ^the inverse Fourier transform of (A. 13) (with ^{the aid of}

Ref. [27] ^{for some} integrals), ^{and we} finally ^{have :}

This is the solution reported ⁱⁿ equation (22),

References

[1] ^{Colloides et} interfaces (Les Editions de Physique, Aussois) ^1983.

[2] PIERA0143SKI, P., Contemp. Phys. 24, ^{no. 1} (1983) ^25-73.

[3] CRANDALL, ^{R. S.} and WILLIAMS, R., ^Science 198

(1977) ^293.

[4] OHTSUKI, T., MITAKU, ^{S. and} OKANO, K., Japan. ^J.

Appl. Phys. ¹⁷ (1978) ^627.

[5] JOANICOT, M., JORAND, M., PIERA0143SKI, ^{P. and} ROTHEN, F., ^J. Physique ⁴⁵ (1984) ^1413.

[6] LINDSAY, ^{H. M. and} CHAIKIN, ^P. M., ^{J. Chem.} Phys.

76 (7) (1982) ^3774.

[7] MITAKU, S., OHTSUKI, ^{T. and} OKANO, K., Japan. ^J.

Appl. Phys. 19 (1980) ^439.

[8] MITAKU, S., OHTSUKI, K., HIRAKAWA, K., HANDA, ^H.

and OKANO, K., ^{J. Fac.} Eng. ^Univ. Tokyo (B)

XXXIV (1978) ^605.

[9] BIRD, R., ARMSTRONG, R. and HASSAGER, O., Dynamics of Polymeric ^Fluids (ed. ^John Wiley & Sons) 1977, ^{vol. 1.}

[10] LINDSAY, ^{M. and} CHAIKIN, P., ^J. Phys. Colloq., ⁴⁶ (1985) C3-269-280.

[11] Physics Today, ^{37 no. 1} (1984).

[12] CHAIKIN, P., DOZIER, ^{W. and} WEITZ, D., ^Periodic Structures in Driven Colloidal Crystals, ^J. Physi-

que Colloq. ⁴⁶ (1985).

[13] HIRTH, ^{J. and} LOTHE, J., Theory of Dislocations, (ed.

McGraw-Hill) ^1968.

[14] ACKERSON, ^{B. and} CLARK, N., Phys. Rev. Lett. 46, no. 2 (1981) 123.

[15] This interaction decreases as d-2 where d ~ 20 a is the

nearer dislocation distance while the self dissipated

power is r-2c where rc ~ a is the core radius [12]

of the dislocation.

[16] LANDAU, ^{L. and} LIFCHITZ, E., Théorie de l’Elasticité, (Ed. Mir) ^1967.

[17] DUBOIS-VIOLETTE, E., PIERA0143SKI, P., ROTHEN, ^{F. and} STRZELECKI, L., J. Physique ⁴¹ (1980) ^369.

[18] ^{LANDAU, L. and} LIFCHITZ, E., Mécanique ^{des Fluides} (éd. Mir) ^1971.

[19] LUND, F., Phys. Rev. Lett. 54 no.1 (1985) ^14.

[20] ^Handbook of Mathematical Functions, ^{ed. M. Abramo-} vitz and I. Stegun (Dover Public., ^New York)

1970.

[21] FRIEDEL, J., Dislocations (Pergamon Press) ^1964.

[22] LEIBFRIED, G., ^Z. Phys. 127 (1950) ^344.

[23] ^{LOTHE, J., J.} Appl. Phys. ³³ (1962) ^2116.

[24] ^JORAND, M., ROTHEN, ^{F. and} PIERA0143SKI, P., ^Disloca- tion motion ⁱⁿ colloidal crystals, ^J. Physique Colloq. ⁴⁶ (1985).

[25] ^SCHWARTZ, L., ^Méthodes Mathématiques pour les Sciences Physiques (Hermann, Paris) ^1961.

[26] MATHEWS, ^{J. and} WALKER, R., Mathematical Methods

of Physics, 2nd edition (ed. ^{W. A.} Benjamin Inc.)

1970.

[27] ^Table of ^Series, Integrals ^{and Products, ed. I. Grad-} shteyn ^{and I.} Ryzhik (Academic Press, ^4th ed.)

1980.