Experiments on 2-D suspensions

Advances

in Colioid

and Interface

Science.

17 (1982)

299-305

Elsevier Scientific Publishing Company, Amsterdam -Printed

EXFERIMEWi_;

ON 2-D

299

in The Netherlands

SUSPENSIONSf

3-L. BDUILLOT, C_ CAMOIN, FI.BELZONS, R. BLANC ARD E. GUYONP'L Departement de Physique des Systgmes Desordonnes, Universite de Provence-Centre de St Jerome, 13397 Narseille, Cedex 13, France

CONTENTS I. ABSTRACT -..................._~____._._____..._______.______.__.._.____2g 9 IWRODUCTION .......................................................... zss If. III. 0 EXPER~~~E~~Tf~L &~NDIT~ONS ...........__....._.________________.__..._____30 IV. .._._......~___.__._______~______.._.________3V V VELOCITY PIWFILE ......... ..___301 1’. VISCOSITY OF ZD SUSPENSIONS ...................................... VI. 2 STATISTICAL STUDY OF CLUSTERS ..............._._.-____-___._.__________30 ..__304 VII. ACKNOG!LEDGE14ENT ................................................... .._....~....._....~_._.~_....._.~~....__30 4 "III. REFERENCES .................... I.

ABSTRACT The rheo7ogical properties (velocity profile, viscosity) and the statistical distribution of objects in quasi-typo-dimensional suspension of macroscopic spheres submitted to a shear is reported. The statistical data are analyzed by analogy with percolation_ II.

INTRODUCTION De Gennes (ref. I) recently suggested that plug flow, as observed by Cox and Flason(ref_ 2), could be analyzed by analogy with percolation_ A key point, from this point 0";view, is that hydrodynamic interactions between particles al‘lObJ the formation of non-permanent clusters. The mean size of these clusters increases an infinite with the volume concentration t_ For a critical concentration 2 c' cluster appears_ Above it the suspension has a two-fluid hydrodynamic character: a single phase composed 3f the fluid and the finite clusters (the "soup") below the threshold, and above it this former phase and a new one made of an infinite cluster. Our aim is to test this concept using a statistical study of suspensions: Z) for a given concentration, we obtained the mean size and nu:nberof ciusters, the +Supported in part by DRET. .. "Permanent address: Ecele Superieure de Physique et de Chimie Indtistrielle,10 rue Yauquefin, 75231, Cedex 5, Paris, France. OOOl-8686/82/0000_00oojso1.?5

0

1932

Elsetier

Scientific

Publishing Company

300 fiuctuations with time of these parameters, and the shape of clusters. etc_; 2) when I increased, the modifications of these parameters were followed; 3) the :rheologicaibehavior of suspensions (viscosity) were compared with the statistical data. Some interesting conclusions have been reached. III_

EXPERIMENTAL

CONDITIONS

facilitate this work, we have used suspensions with particles in the same plane, i.e. "quasi-two-dimensional"suspensions, using the following procedure: cm a heavy liquid. a thin layer of a lighter liquid was deposited, the density of which has been adjusted to that of the particles (spheres 5 mm in diameter)_ The To

thickness

of

the

light

liquid

is

equal

to

one

sphere

diameter.

The concentration may be defined following two different ways: 1) a volume concentration .I.i-e_ the fraction of the total volume of the layer occupied by the spheres; 2) a superficial concentration .F~:in the middle plane of the layer which intersects equatorially the spheres, :,sis the fraction of area occupied by the sections of the spheres_ These two concentrations are related by 3 = 32 ---. Both a statistical and a rheological studyswere performed on quasi-two-dimensional suspensions undergoing a shear in a Couette viscosimeter with the inner cylinder {radius Ri= 17 mm or 33.5 mm) at rest. The angular ve7ocity of the outer cylinder -1 (radius Re = 92 mm or i8D mm) is of the order of 0.1 s _ The Reynolds number Re is between 0.03 and 0.2_ IV.

VELOCITY

PROFILE

Iieasurementswere taken of the angular velocity of the spheres versus the radius r of their circular trajectories and concentration_ Fig. 1 shows several velocity the experimental velocity of profiles_ One can see that when cs is befob 0.4,

01 0

2

4

Fig. 1. Relative a sphere.

5

velocity

8

10

profile

r-

versus the radius of the circular trajectory of

301 the sphere agrees quite well with

the

theoretical

where r and 4 are the polar coordinates on

the

geometry

of the

viscosimeter;

of

and

the

one

for

a homogeneous

A and 3 depend

sphe-re Centers;

C! is the

angular

liquid:

velocity

of

the

only

external

cylinder_

When ss is larger than 0.4, the velocity profile has a tendency to flatten except near the inner cylinder where velocity goes down to zero rapidly. A larger part

of the

suspension

has

a solid-like

behavior

as

in the

plug

flow

profile

ob-

served by Cox and Mason (ref. 2) in a Toiseuille's geometry_ VISCOSITY OF Xl SUSPENSIONS From the experimental determination of torque on the inner cylinder, we define an "effective" viscos'ty n; that is, the viscosity of a pure liquid which, in the same experimental conditions, would exert the same torque on the inner cylinder. ‘I _

if no nr

is the

viscosity

of the

layer

of

the

pure

light

liquid,

the

relative

viscosity

is:

We observed that: 1) for low concentrations, the relative viscosity nr follows the law:

where the experimenta. values of the constants :Z (~-~2 and pLs (~~~1.3) agree well with an Einstein-like calculation (ref. 3,4) which we obtained for two-dimensional suspensions; x = 2 and 1 = 4/3 (recall ri= 512 for 30 suspensions); Pj for larger S

concentrations (3s>O_1), this last descript 3n does not apply since it does not take into account hydrodynamic interactions. Our statistical study was conducted in this range of average concentrations_ Fig. 2 shows that n,,. diverges when
E

..

.

s

-9” 0

1

2

3

-4

.5

.6

-7

.8

_9

Q

Fig.

2. Relative viscosity versus concentration c for different values of the _I angular velocity of the external cylinder in the CT ouette experiment. 0 :?= 0.177 s ; 0 r.= 0_0037 s-l; * !?= 0.0282 s-1; V = Eilers-like lakJI 413

?I.

STATISTICAL STUDY DF CLUSTERS When one observes a suspension in a shear, one can see that spheres may be isolated or grouped in clusters. It is known (cf. Ratchelar and Green, ref. 6, or Van de Ven and Eason, ref. 7) that in a suspension submitted to a shear, a simple pair of spheres cannot come in true contact under the action of the hydrodynamic forces only and there is a minimum distance between spheres. Hovever, in a real suspension, due to multiple interactions between spheres, the situation is quite complicated and the question of the existence of true contacts is still open (see De Gennes, ref. lb). Accordingly, we made the following convention: we state that two spheres belong to the same cluster when their distance is lower than the resolution of the method of observation (photographic examination); that is, about 0-l radius of sphere_ Our statistical study of the clusters versus the concentration 3s was conducted. The first results are given here. If s is the number of spheres in a given cluster and if ns is the number of clusters of s spheres, the following parameters currently used in percolation (see D. Stauffer, ref_ 8) can be calculated: 1) the average number of spheres in a cluster is defined as:

303 XSll

s1

=-.rs--z

N

$n=q=

<5'

,

S

Iarhere N is the total number of spheres in the finite size suspension and Nc is the total number of clusters; 2) S is another definition of the mean size of a cluster 2 (mass average), which privileges the largest clusters:

in the theory of percolation, the singu7ar behavior of S2 around the percolation threshold is the analog of the magnetic susceptibility at the Curie point in a conventional phase transition. Fig. 3 shows the value of S1, together with the relative viscosity, versus the concentration .3s- One can see that as long as I is smaller than 0.6, the two curves have the same relative variations. For lzrger values of ls, SI appears to increase more rapidly than the relative viscosity_

.*

-

0 0

.5

(D,

Fig. 3. Relative viscosity I+ and average number of spheres in a cluster S1 versus the concentration os.

304

For 61s01 z the order of 0.65, a very large cluster appears which includes a large fraction (70 to 80%) of spheres and completely surrounds the inner cylinder_ In order to display, in a more dramatic fashion, the statistical behavior of the clusters in the range of concentrations around O-6-0-7, one can use the parameters Si and Si which have the same definition as SI and Sz but with the largest cluster subtracted.Figs.4a and 4b show the variations of Si and S; versus +,_ One can see that there is a maximum for these two parameters in the range 0.65 -0-70 of ~~,which is an indication that a critical behavior could take place in 2D suspensions. However, further studies are needed in order to check this last point. Specifically, the number of spheres (%lOOO currently) must be signficantly increased if one wishes to break away from size effects and wall effects.

_,t

50_

6

(a)

(b) 40_

30_

20_

lo_

+ k 0

01 1

-5

Figs. 4a and 46.

.6

-7

'D,

.4

.5

.6

.7

'D,

Variations of Si and S,jversus the concentration 9,-

VII_ ACKNOWLEDGEMENT We are grateful for stimuiating discussions with D. Stauffer. VIII_ 1

2 3

REFERENCES

P.G. De Gennes, J. Physique, 40(1979)783; P_C_H_M_ III. Madrid (lg30)_ R-G. Cox and S-G. Mason, Ann. Rev. Fluid Mech.. 3(1971)291. A. Einstein, Annalen der Physik, 4(19)(1906)289;corrections, ibid., 34(I9II)59I; investigations on the theory of Brownian movement, edited with notes, R- Furth, ed., Dover Publications, Inc., New York, 1956.

305

4

5 5 7 8

H, Belzons, R, B‘lanc,J-L- Bouillot

and C. Camoin, C-R. Acad. Sci, Paris, 292 II (1981)939. R. Pltzo'Id,Rheo7. Acta-, 19(1980)322. G.K. Batchelor and J, T. Green, J..Fluid Net%., 56(1972)375_ T.G,% Van de Ven and S-G. Mason, J. Colloid Interface Sci., 57(1975)506. D_ Stauffer, Phys. Rept.. 54(1979)1.